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The angle of the front wheel would suggest that this is more than a broken driveshaft and could be a can of worms. It appears that the wheel has taken the brunt of the collision - it may have been turned out on partial lock whilst the car was parked. You would have needed expert advice before considering a repair settlement. The devil is of course in the detail - but if you had fully comp insurance - your insurance company is in breach of their obligations. They cannot wash their hands of the matter and should have acted in your best interests. As a minimum, the car should have been recovered to their approved repairer (or one of your choice) for a full evaluation. It's not good enough to leave you to deal with the third party - infact a decent insurance company would not want you to enter into direct debate with the third party or their insurers. Was this a known insurance company or a Lloyds syndicate?

If you have not yet accepted their offer, you still own the car, and you would not be buying it from them, just retaining it. plus receiving a cash sum to compensate you for the damage. BUT, you will bear the risk that the repairs could turn out very expensive. If you have accepted the offer, ownership passes to the insurers, and they may have already sold it.

If it's written off as a total loss, you can ask the insurers to let you have it, with an adjustment to the payment they give you. If you argue the amount of their offer, they may even give it to you for little or nothing. I've heard of that happening. Then its up to you to have it repaired. You may find that, after repair, insurers will only give you 3rd Party cover.

If the claim has been settled on a total loss basis - the car will be declared Car N or maybe Cat S - as such, it wouldn’t be an attractive repair prospect for anyone.