Maybe look up north for paralegal work as a lot of firms have set up dedicated transaction offices, I know addleshaw Goddard, fresh fields, lovells all have offices in Birmingham or Manchester where you could cut your teeth. Seeing as you have an LPC as well you would probably get a position and with AG they actually recruit 8 trainees per year from their pool or paralegals, so whilst you might have to do a short stint there it might pay off in the long term. Alternatively go to an agency and as you have LPC you should be able to find somewhere in London