I did something similar to this for a year. The round trip was 58 miles, not 60, but it had some decent hills so it involved close to four hours per day on the bike. I did it five days a week to start with, but backed off to four and, sometimes, three days.
Whether or not I would recommend it depends on your lifestyle and what you want to accomplish. From the sound of things you have plenty going on, and I strongly suspect that you would find this investment of time every day detracted from either school or family life. Plus, twenty hours per week is actually more than you need to train in order to get super-fit, and going hard on five successive days, week after week, probably isn't even the best training strategy.
I'd suggest doing it on alternate days, so maybe three times per week. Assuming that you want to ride at least once on the weekend, that's probably closer to optimum training-wise, as well as slightly less onerous time-wise.
FWIW, the commute I did for longest was a 32-mile round trip. That seemed to me to be pretty much ideal, took only about two hours out of my day (less when you net off the time I would have been driving), gave me enough training volume to stay in good shape, kept me pretty cheerful.