I can pretty much jog comfortably at 5 miles an hour for somewhere between 30 and 55 minutes, depending on the day. I do, however, get bored, and I feel like I'm not really challenging myself, so recently my plan has looked more like 10 minutes at 5 miles an hour, 5 minutes at six miles an hour, walk a few minutes, 15 minutes at 5 miles an hour, 5 minutes at six miles an hour, and I'm done. By that point I am pretty much entirely exhausted. My question is, is increasing the speed for a portion until I find that I need a break partway through less beneficial to increasing endurance than just running as long as I can at a lower speed?

Basically, your question is that of low intensity versus intervals. IMO there is a place for both in a cardio routine. Lower intensity/longer duration does burn a higher percentage of fat and allows for endurance training. Higher intensity/shorter duration burns more calories and gives other cv benefits.

Mix it up- one is not 'better' than the other; it all depends on your goals. You can also have fun mixing up the intervals. Try fartlek running- where you run at all kinds of intensities at random, from sprinting to comfortable to really slow to medium to high etc.