Not signing the terms and conditions is largely irrelevant - you are deemed to have accepted them by using the solicitor's services.
Unless you have anything in writing it is going to be very difficult for you to prove that the other man should pay the bill.
You could have a case against your friend, using the doctrine of estoppel, but it is not guaranteed you would win and will incur more legal fees.
This bill will have to be paid - the solicitor will not hesitate to enforce it if you don't pay. Ask him for time to pay whilst you try and sort it out. See if you can come to an agreement with your friend about sharing the cost.
I can't understand why your friend was so keen for you to take the former employer to court. An industrial tribunal would be as effective and there is no need for a solicitor.