Is your laptop not covered by a warranty? When did you purchase it?
How has your laptop broken? Is it repairable? You might be able to repair it or buy a replacement and claim through the Access to Learning Fund.
Does your library or disability service provide a laptop loan scheme? It's worth checking as some do. It's not an ideal situation but if you can borrow a laptop until you have a replacement it's better than nothing.
As for DSA, was getting a laptop discussed in your Needs Assessment and, if so, was it stated on your report? You said you were able to claim one but I just want to make sure. If this wasn't mentioned on your final report I think you'll need to get it amended and, even if you can get it amended, the process may take a few weeks.
There's no way of "fast tracking" things as far as I'm aware though it is worth speaking to your disability service. I suggest contacting both your disability service and funding body as soon as possible.
But please don't think you'll "utterly fail" your exams. It will work out one way or another.
You can't 'claim' a laptop, but one may have been recommended for your use during the course. You should have been advised to to purchase warranty for your laptop, the cost for which would be through your DSA. Go back to your assessor and speak to them, they may be able to recommend it be replaced it, but it depends on your funding body.