Sorry to hear of the issue.
There are a few issues here. The family say they have no money so whilst you could sue for the 4K, recoverability may be an issue. The other issue is the insurer will not help and neither will the Motor Insurer's Bureau (MIB) from what you say.
I don't know if you have notified your own insurer - if you have insurance, your insurer will say you need to report this to them. If you have cover, they should deal with the repairs.
If you prefer to sue the family of the child, you don't need a solicitor - as it's a district court matter and you can do this yourself.
You need a letter before action and if they refuse to pay, issue a summons.
The following website has both - the letter and summons, and they also offer a solicitor-check service too :
You could go to a law firm but costs will come in to this - the above site is a cost-effective way to pursue a claim like this yourself, but with a lawyer who can check the documents to ensure they are correct before you pursue court action.