Since you live in a neighborhood where most everyone looks out for each other, why not reach out to the others and see if they will help you with the sidewalk through this ridiculous winter. Maybe you can respond in some other way despite your physical issues - babysit their kids so they can have a Saturday night date, make them a nice meal, or whatever you can do using your time and talents but not your money. Avoid the problem neighbor and let her see your true friends helping you out.
The true friends are the ones who should be using your pool! In fact, you can make them a "gift certificate" for helping you out with the snow, inviting them to a pool party or giving them 1 or 2 free passes to the pool at the time of their choosing. If someone in your neighborhood has a snowblower, perhaps that person could do the sidewalk, which would clear it and keep the ice from forming to begin with. You can pay them back in the summer with the thing you have - a pool - and not have to pay any money.
Meantime, I would stop inviting her over or allowing her to drop by - just say you have a herniated disk and you are in too much pain to be a proper hostess. When the good weather comes (which I hope it will!), put a lock on the gate to your pool (which I assume you have for safety reasons) and stop allowing this ingrate family to use it. Just say there are insurance/safety issues and you are not healthy enough to be a lifeguard or to clean the pool regularly, so you are unable to be as generous as you have been in the past. Your husband isn't around much because he took a 2nd job to pay off the medical bills (no shame in that). Don't ask for sympathy, just state the facts even in the form of "As you know, I've been quite ill for 2 years and in a lot of pain, and John has had to pick up the slack."
She's not your friend. She has a lot of problems obviously because she goes through life complaining and criticizing others in order to feel better about herself. Pity her, but don't spend a lot of time on her. Don't get into accusing her of reporting anonymously - it doesn't help to have her know that anonymous isn't anonymous in your town! It will just get her going and perhaps motivate her to do something else nasty. Wasting too much energy on finding out who called in your sidewalk is keeping you from finding a creative solution to your problem.
Move on. She's got problems and it's not up to you to solve them.