Well given the sod patch is an original technique I deveoped, I'm not sure how you did that idea. Fake grass mats do not work Since you say you tried the crate training, how large was the crate? Did she eliminate in the crate every day? how long after drinking does she urinate? How long after eating does she defecate? When do you feed her? Again, what cleaning supply are you using and how are you using it?
The ideas and techniques I've described are proven to work so it sounds like you may have not been consistent in how you executed them or didn't do them long enough to see improvement. The log book works wonders. Now if the problem is that she eliminates each time you leave it could be separation anxiety and not a house training issue. For that I'd recommend a DAP collar to help reduce anxiety and desensitizing training. Dap collars are proven effective for anxiety and work for soiling due to anxiety.
Practice putting her in the crate, leaving the house, opening the door immediately and rewarding her with a hot dog treat if se did not eliminate.. This teaches her that you leave but come back quickly. Once she seems to not do anything when you initially leave, lengthen the time she must not eliminate for you to come back in. Change the time as well. Make it 2 minutes one time and 10 mintues another, so she never knows if you are gone for an hour or gone for 2 minutes. It helps her stay calm for longer periods of time, just be sure you reward her when she does not eliminate
Another thing that helps is to do things that might make the dog feel you are leaving and then don't such as putting on your coat or picking up your keys. Or leave without doing those things. This helps remove things that might trigger the dog to become anxious.
These should help her separation anxiety and help stop her eliminating if it related to anxiety. It will not be an overnight cure and will take work on your and your family’s part to be consistent in your interaction with him. Here is a site that also offers idea to combat separation anxiety.
You will need to clean everywhere exceptionally well using lots of enzymatic cleaner and let it soak in to remove the odor. Personally in over 40 years i've never seen a dog not be able to be crate trained unless there was a medical issue, so I would get her checked over. Then try some of these methods for pad training but no matter what do not let her run loose.
Here are three pad methods that I already have written up even though I don't recommend pad training for reasons I mentioned earlier.
In the area that you want her to use the pad, set up an enclosure. You can use child gates, free standing pet enclosures, or anything that will serve to enclose the area you want her to use. These are the times when you will be taking your dog to the enclosure. When you feed or give your dog water, first thing in the morning, last thing at night and after extended play times.
You will want to start this when you are not working or when you can spend a couple of hours in the morning and evening exclusively doing this. Enclose or barricade the area and put pads over the entire area. Scent one of the pads and place it where you will leave it when there is only one pad. When you and your dog get up in the morning, instead of taking her outside, take her to the enclosed area and put her inside it. You can leave her a few toys but basically put her in the enclosed area. It is best if you can stay there and keep an eye on her. Do not engage her with talk or play or reassurances. Just be there so you can see her when he actually goes. She may be reluctant, but do not cave in to whines or cries. You must remain calm and in control and ignore her. Sit in a chair and read or do some other activity to keep you occupied while she is in there so you are not focused on her but can still see her. It may take some time, especially the first time, but she will eventually go on the pads as she has no where else to go. When you see her start to go, you can say in a low calm voice “good potty”. It needs to be in a low quiet voice so she isn’t startled into stopping. Give her lots of praise for succeeding and perhaps a little treat and you can then remove her from the enclosure and open it up so she has access if she want to use it. The first time may take an hour or more, but if you do it first thing in the morning, it is likely to take less time.
Repeat this according to the circumstances I mentioned earlier concerning when she should be going and a few extra times during the day to try and avoid accidents. After a few times going in the enclosed space and being praised for it and getting treats she should not be as reluctant to go in the enclosed area. Continue to praise and say the key wood “Potty”. When she reaches the point where she is going to the bathroom almost immediately in the enclosure on the pads, you will want to start taking up some of the pads so only a portion of the enclosed area has pads. Repeat as if the whole area was covered. Your dog should go on the pads and not the floor. If she does, continue the praise and treats. Slowly reduce the number of pads in the area until you are down to one or two only. If she should happen to miss the pad or go off the pad, add some more for the next time as you probably were taking them away too fast. At this point she knows the pads are where she should be going.
Once she is going on the pads and you are down to one or two pads, don’t close up the enclosure when you put her in it. He should still use the pad as she is now used to it. When she does, again praise, praise and more praise, let her come out the opening. Continue this process if it is necessary, but with the enclosure being open, you may now find that he is going there on his own. Once you notice she is using it without you putting her in there, you should be able to take away the enclosure or barrier. She should now be trained to use the pad.
Depending on how quick a learner she is, you may find she starts going in and using the pad early. If so the barrier can go earlier, but you should still only slowly reduce the number of pads she has to go on, so she doesn't miss, use the floor and think it is ok.
Second method is similar but you would leave her in the enclosed area all the time unless you are taking her for a walk, and just slowly removing paper and decreasing the area inside the enclosure that is papered. As long as she continues to use the paper, you are not moving too fast. Once you are down to a small area, she should be trained. I suggest this because while the dog is contained in an area with paper, the dog may not be as stressed as when there is a cage all around him.
Set up a contained small area with the paper. Then you will have the dog on a leash attached to you. I attach the leash to a belt loop on my pants. This way she is always where you can observe her easily and stop her before she starts urinating and place her in the potty area. When you see her start to squat, give a quick short tug on the leash and in a low toned, firm voice say "NO". Do not be upset. Pick her up and put her in the potty area. With this method it is best to just stand outside the area looking around ignoring the dog until she goes. Remember praise is everything.
If none of these work, you will have to crate her when you are not home and leash her to you when you are home. If she is leashed to you, she won't be eliminating without you seeing. If you can see her, you can give a correction before she eliminates and take her outside. If she is in a properly sized crate and she eliminates in it, she will be forced to sit in it. They don't like that. No padding in the crate. Bare tray only. If it has a pad, it will soak up the urine and it won't be uncomfortable. it has to be uncomfortable for her or she won't learn. I know it will be hard on you as you will need to rinse or bathe her when she eliminates in the crate, but she won't be eliminating on your carpets.