A wealthy friend of mine was in a similar predicament a couple of years ago , I think he had six points already and was caught at 120+.
He hired a legal team from London who specialise in speeding cases and argued that he needed his car for his job, his kids would suffer etc...... End result, he got banned for 6 months, had a £2500 fine and still had 12 points on his licence when he got it back. The legal bill was around £7000 ( he never told me the exact amount) .
We all speed at some point and you got caught , take the punishment and don't waste a load of money on a solicitor as I think the end result will be the same.