That makes no difference, he's claiming from the other party who have admitted liability. If he was comprehensive on the shed, he would still be claiming from the other party. The issue is the lack of support from his own insurers and their subs, an area which, no doubt, they made great plays on to generate and secure business in the first place. Like you said, "why bother". Once they have your money they're like rats on a sinking ship.
I would suggest that you put the onus on the other parties insurers (which you would have expected Admiral to do for you) to sort out your car. They have a claim against them, tell them in no uncertain terms to sort it. It's up to them to inspect the vehicle to decide whether repair is viable. If, as you suspect, they will consider it a total loss, make sure you only accept market value or better, you're not obliged to accept their first offer. As Scuba suggested, if you have any legal services attached to any other policy's, memberships, credit card's, home insurance, give them a call for advice, or even the citizens advice bureau. Make sure you keep records of all correspondance etc.
Whilst the other party has probably just called his claims hotline to sort everything, it is unlikely that his broker will be interested, they would just refer you back to your own insurer, or at best, the other parties actual insurance company. At least you can tell Admiral to get stuffed at renewal.
Either way, you have a legitimate claim, stay cool, and don't let the b@stards grind you down. Best of luck.
Great advice
Being TPFT in this instance is irrelevant. The other party are at fault and have admitted liability, therefore the only obligation you have to your own insurance is to inform them of the incident. In an ideal world they would take over the repair of your car and provide a hire car (this is how they make money out of accidents that are not their clients fault ) and then claim it all back from the other drivers insurance.
If the other party has a hire car already then there is a claim in the system already so you need to phone them and not take no for an answer!!
PS: The broker won't want to know on Monday, why would they? They source policies not provide claims advice/help.