For you ticket, you should absolutely seek counsel based on what you have told us in this post. That attorney should work in the area of traffic accident defense matters.
As for the city paying you, they should, but that means nothing. You must prove to the city that their driver was at fault in order to have a claim paid, and that will be very difficult to do without a different sort of attorney -- a civil litigator. In order to pursue the city government, there are quite likely very strict timing guidelines for bringing claims so you need that attorney to help you if you believe the property damage is so significant that it is worth pursuing.
Finally, if you have collision insurance, let your carrier know and they will take care of your damage if you choose not to pursue the government.
In no way am I offering you legal advice, and in no way has my comment created an attorney-client relationship. You are not to rely upon my note above in any way, but insted need to sit down with counsel and share all relevant facts before receiving fully-informed legal advice. If you want to be completely sure of your rights, you must sit down with an experienced criminal defense attorney to be fully aware of your rights.
With both parties being given tickets it appears you have a situation where the police officer did not know who was to blame so he ticketed both of you to let the court figure it out. As stated you should hire a traffic ticket attorney. Your goal is to defeat your ticket. If you testify against the other driver and his found at fault then you have established liability. Then you have an argument that your damages should be repaired. However right now it appears as if both parties are at fault and in those situations everyone pays for their own damages.
Deal with the tickets and then based upon the results you can deal with the property damage.
This response is for general information only, no attorney client privilege has been established by this communication.
Sign up to receive a 3-part series of useful information and advice about personal injury law.