Am I the only one who thinks a 24 mile commute on 'just a cheap one' is a helluvan ambitious target for a newbie? No offense, but 50 miles a day is caning it, even with a good bike and some serious miles under your belt.
If it hasn't arrived yet, I'd think seriously about revising your choice of steed. If you're going to do that kind of mileage (and you can, there's nothing inherently undoable about it), and especially if you plan on actually enjoying it, having a good bike will make an enormous difference.
I'd recommend either upping your budget (taking into account all the money you'll be saving by not driving) or looking for a good second hand bike.
Oh, and, personally, about seven and a half miles, 25 mins.