Agree you need to contact a solicitor who specialises in this. Also where abouts are you as the rules on no win/no fee etc differ across different areas of UK. For example, I used to practice in NI and there is no win no fee there.
On the face of it, if the driver is found guilty there should be no dispute of liability and you may have a good claim. A solicitor should write to the drivers insurers and set your daughter up with a consultant psychiatrist appointment. The psych will write a report on the impact of the accident on her, and this will be used to talk money with the insurers. It's quite usual to go back and forth on offers two or three times, but your solicitor will be able to guide you on whether you should accept an offer or not.
It is possible to settle a case involving a minor outside of court, but there will be a court date in which the judge will have to approve the settlement, to ensure best interests of the child. But if you have agreed a settlement and just looking for judicial approval, your daughter should not have to give evidence etc. Depending on the extent of psychiatric injuries, and your jurisdiction, this will either be county or high court.
If you are successful in your claim, the solicitors should recoup costs from the other side.
I would contact a personal injury firm that's local to you, not a big chain.
Also keep in mind a lot of solicitors now do the majority of these cases over the phone, it's much more efficient for everyone and instructions don't really need to be taken in person. If you want to meet in person they should accommodate this so do say if that's the case.