Who is your insurer? Is it fully comp? legal fees included?
Either way, go see a soloicitor too, you might end up getting properly shafted otherwise. The other driver has clearly been advised to shut up. Act quickly, there may be deadlines that you are not aware of for claims and defences
Just tell the truth, keep your notes, particularly the admission, and hope for the best. Do not trust anyone's word, only signed written documents.
I really hope it works out for you.