Hi, this is Dr. Joey. I am so sorry I was unavailable when you posted your question. And, that no other expert jumped in to answer you.
Sounds like so far you have done a great job. The first step is to stop the bleeding. If there is a remnant part of the nail that is loose/dangling, then you will need to remove this. This may be difficult depending on how torn/loose the nail is. My hope is the broken part of the nail is gone. Hopefully, all bleeding has stopped entirely by now. If the bleeding has not stopped, you can go to the pet store this morning and get QuikStop or another stop-bleeding product to help. As well cold (ice) helps.
If the paw is swollen you may soak the paw in a warm solution of diluted betadine (dilute with water until is a tea color); you can purchase this at almost any pharmacy over-the-counter. This helps with infection. The warmth draws out fluid.
Beyond this, there is not a lot you can do at home. IF there is a piece of nail which is very firmly adhered at the base, but obviously broken, then a veterinarian may need to remove, which might require sedation since this can be very painful. As well, pain relief medication may also be indicated. I wish I had an easy over-the-counter pain relief option for you but there are none that are effective and safe. You canNOT use ibuprofen, acetaminophen or naproxen. You might be able to use aspirin (baby aspirin) at one-half every 12 hours BUT I always warn my clients with aspirin I find it does not work very well to provide pain relief, and there is a risk whenever this particular product is used of it causing a stomach ulcer. So, you may take that information and do what you feel is best for your dog.
Please keep me posted. I am always happy to answer followup questions. You can reply to this question at any time to submit a followup. My goal is to provide you with excellent service today. If there’s anything else I can do, let me know – it’s been a pleasure working with you and I hope to assist you again soon! Kindly submit a rating as you exit our chat.