Hello,
Your best bet for finding a local mechanic would be to ask some friends for a trustworthy garage or to check the yellow pages for your area. Here at justanswer we are from all over and your best bet would be word of mouth references.
Hopefully if you did make a mistake with the positive and negative no damage to control modules have occured. By your description, it sounds like the starter solenoid might be sticking (that is the clicking you hear). Most likely the starter is the problem but make sure you have secured the neg and positive terminals. A loose connection can cause a voltage drop and prevent the starter from turning. If they are tight most likely it is a starter that needs replacment. ---Another remote possibility is that the engine is seized in some manner but most likely not---This should be able to be replaced by any repair shop and it will save you a lot of money on labor...one suggestion is to ask or make sure the repair shop uses BMW parts and not non OEM parts as you will get much more life from an original BMW part.
I would think the starter would need to be replaced. The cost of a starter will vary on location but I think it will be aprox 150-200 dollars for the part. If you do decide to go generic it could be less.The time to install a starter would be aprox. 2.0-2.5 hours depending on the condition of the exhaust...rusted bolts and studs would have to be replaced and could cost a bit more if applicable. If you brough it do the dealer you will have expert installers and a warrenty for unlimited miles for two years on parts and labor but the labor for this will be twice as expensive than a garage.
If control modules where damaged it could be very expensive to fix. They can be replaced but the cost might not be worth the value of the vehicle depending on the extent of damage. But since you hear that "click" when turning the key that leads me to believe that the starter is trying to turn the engine but the starter is not functioning. If you heard nothing, while this could still be a bad starter that could mean something else like a control module was damage. It is better to keep it simple and the simple things are most often the problem. The starter is the logical first step. Thank you.
BMW Mechanic
Level 1 BMW Master Certification, 9 Years Dealer experience, Team Leader, Shop Foreman