Yes, he does.
The 'clever' part of that problem comes from the fact that having completed one lap at 30mph he will have taken as long as he would have needed to have completed both laps in, ergo the second lap is irrelevant as he's already hit the time limit after one lap.
However what you actually asked was just what speed would he need to average 60mph, which would be 90mph, hence my comment that I think you phrased your question wrong and didn't actually ask what I think you intended to ask.
All you asked was what speed would he need to do on the second lap to average 60mph over both, the answer to which is 90mph. Had you asked 'to qualify' then obviously, there is no answer as he is already out of time and cannot qualify.