1. Try to go through your insurance company first because this sounds like a personal injury claim as part of the original incident - assuming you have personal injury cover under your policy.
2. If you choose to continue with the law firm you've instructed you need to read their terms closely. Make sure that they can only charge you a percentage of whatever you win from the other side. This is typical in no win-no fee engagements - they take a portion of your damages but crucially you're not expected to fund their costs above and beyond the total value of the damages recovered from the other party.
The idea is that their costs come out of your winnings, so to speak, and they will (should) be keeping a close eye on their costs versus the likely payout they can claim against. Ideally I'd be looking for them to agree a percentage of payout rather than a fixed fee or "per hour" cost (which could exceed any payment you win from the other party).
If they won't agree to cover their costs out of the likely payment from the other party then that may be an indication that they feel the chances of winning are slim but not zero, so they may be trying to cover their bases by saying "no win-no fee... but if we burn through more 'per hour' costs winning for you than the payout will cover then we want you to cover the shortfall". If this is the case then I, personally, would find a different firm to work with.
[Speaking as a trained, but not practicing, solicitor.]