It all really depends on what you had agreed with the plumber before the work began. Presumably you had agreed nothing in writing so you have only each other's words to reply on. You clearly accept that the copst for the parts supplied are fair and reasonable and you have ineed paid the sum of �310 in that regard. The main dispute therefore seems to be over the payment for the time spent by the plumber in attempting to fix the problem.
Now... the question arises: if the problem with your boiler had been fixed to your satisfaction would you have paid the full �700 expected of you ? That seems to be the point at issue, because, as the problem was not fixed you decided that only part payment should be made for the plumber's services.
In the circumstances, I have to conclude that you are onto a "loser" and you should make payment as demanded by plumber. The alternative would be a court hearing which inevitably involves you in extra costs.