You can take legal action. This is best undertaken through a specialist solicitor - some give the first half hour of advice free
You can make a claim to the Criminal Injuries Compensation Authority (UK)(Criminal Injuries Compensation Agency in N. Ireland)
You can seek a criminal compensation order through the courts - which usually comes about through the police bringing charges and the prosecution understanding the extent of your injuries.
If the injuries are serious enough a solicitor is absolutely essential. Remember though that if you claim from an indivdual rather than a scheme and they cannot or will not pay you will spend months and possibly years chasing this down possibly with no satisfactory conclusion.
Go to a solicitor who will tell them what to do next, they will probably be sent to a specialist to assess the injury, and it is an advantage to have been to A+E immediately after an assault (I have successfully claimed after an assault, even though the person who did it was never caught)
You don't need to go to a solicitor, that will cost you money. If in the UK try Citizens Advice. Their advice is free and they will help with Criminal Injuries Compensation (although it does take a long time to come through).