Here's where I taped it. I also taped it inside as well.
Absolutely fine. Your only risk is further breaks if the case is handled roughly.
People commonly travel with cardboard boxes covered with tape!
If you check in for your flight with an extremely by-the-books agent and check that bag, they may note the existing damage on the back of the bag tag and/or make you sign the damage waiver on it. Even then, though, they'll be fine with it.
You'll be absolutely fine. Airports have, to put it simply, seen much worse.
Assuming that, depending on what kind of break is going on there, fragments and/or debris won't come off of it (or harm anyone handling it), it will pass right through without a hitch.
Bear in mind that it'll have other cases/weight on top of it etc during handling/flight, so assuming that bit of tape is fine at keeping it together, no worries.
I once had a suitcase stolen, but not the rest of the luggage.
I put everything in a bin bag, with tape around it, and handed it over like that. The airport-people didn't bat an eye.