If there is no injury, then it's unlikely a solicitor will take on your claim. There's not enough money in it for them.
However, once you have a quote for the repairs to your bike you can simply call or write to the claims department of the lady's insurance company to put in a claim. The driver should have reported the incident anyway and you can quote the police incident number.
As well as the repairs to the bike, you can claim a sum for the hurt and distress that you suffered and any additional costs you have incurred for clothing, transport while your bike is out of action etc. As long as your costs are reasonable the insurance company is unlikely to dispute your claim, since it saves them lawyers fees and other costs.