The new firm will be acting as agents for the Law Society on behalf of the partners of the original firm.
Of course they're allowed to bill for the work that was done for you.
If the original firm's papers were so bad that the agents don't have important documents then they may not even show the actual terms you originally agree to when you instructed them (though I doubt it).
Its highly unlikely that they're trying to extort money from you, and the Law Society doesn't usually instruct incompetent firms as agents
Speak to whichever solicitor has taken over your case and explain your situation to them.
Didn't they explain all of this when they first contacted you anyway?