I've done some work with the Ambulance Service in the past (N.Wales and Surrey). Every time we dealt with a patient, it was logged on a Patient Report Form. This was to protect both the patient and the Ambulance Service and also for patient transfer (medical records) at the hospital. As far as I know, St Johns also do something similar. These records should be kept for a number of years.
The receiving hospital should also have a Patient Transfer Notice, and there should be a log in the A&E department with your details on arrival.
Also, for the Club to say they have no record of the incident, they could have fallen foul of RIDDOR (Reporting of Injuries, Diseases and Dangerous Occurrences Regulations) under the section regarding injuries to a person who is not at work, such as a member of the public, which are caused by an accident at work and which result in the person being taken to hospital from the site for treatment.
The Club are responsible for the safety and security of paying customers. If a contractor is paid by the Club to carry out flooring work, they are also responsible for checking the quality that work and deciding if it comes up to standard. If the contractor had been paid for that work, then someone at the Club must have ultimately decided it was up to standard.
You could always contact the premier League if you don't make any progress with the Club. Also, as others have said, consider another solicitor. It sounds like yours is being lazy by trying to get you to do all the legwork.
Hope you get some closure soon.