The second offence means that the driver held a licence but it wasn't valid for driving the car he was in. The most likely explanation is either
(a) he only held a provisional licence (and wasn't accompanied by a qualified driver) ; or
(b) he only held a motorcycle licence.
Your claim is not against the other driver's insurer; it's against him. (If he's got valid insurance cover in place, he can pass the claim to his insurer. Otherwise he must foot the bill).
The first stage of making the claim is to send a formal demand to the other driver. (The courts won't get involved unless you've already given the guy a chance to pay up). You should send him a letter by recorded delivery (retaining a copy of the letter and the proof of posting). The letter must include:
(a) your name and address ;
(b) the name and address of the recipient ;
(c) the date ;
(d) a statement of the amount demanded ;
(e) the reason for the demand ;
(f) notification that if full payment isn't received within a specified period (I suggest 21 days) you will commence legal proceedings.
If you don't get your money, use the online claims process which is provided by Her Majesty's Courts Service:
https://www.moneyclaim.gov.uk/web/mcol/welcome
Chris