I'm assuming your statement, "If I get myself onto the wrong side of a king verses king, bishop and knight endgame" means that you have the bare king. In that case, you can offer a draw, but there's a good chance that your opponent will decline the offer, either because he knows the mate procedure, or he wants to try anyway. If play continues, your opponent has 50 moves to complete the mate, or it's a draw. (But remember, for over-the-board games, you have to claim the draw for the 50 move rule!)
BTW, it took me about 3 tries before I finally learned the procedure. Having someone show me didn't work for me. I consulted about 6 different endgame books. Most of the book explanations, although correct, were clear as mud to me. Finally, I checked out Pandolfini's Endgame Course. Although a number of people told me not to buy that book because it was TOO basic, it has the clearest explanation of K vs KBN of the books I consulted. The only odd thing about it is that the mating procedure is in 4 parts (lessons 21 through 24), and the lessons are in reverse order. (Don't let that stop you, just start with lesson 24 and progress to 21.)
Another tip about the procedure - Pandolfini's explanation starts at the standard position with the bare king in the "wrong" corner. Pandolfini does not explain the method of driving the bare king from the middle of the board to the edge of the board and into a corner. Unfortunately, as far as I know, there is no set procedure for doing this. You basically have to coordinate the 3 pieces to progressively take squares away from the bare king. The out-of-print book "Chess The Easy Way" by Reuben Fine gives very brief but good guidance on how to try to coordinate the 3 pieces to do this. I'm sure other endgame books also give general guidance for driving the bare king from the center of the board.
Pandolfini's procedure is not perfect, as he doesn't discuss all possible defensive variations, but he does explain the basic procedure very clearly. If you decide to give the procedure another try, I suggest you buy or borrow a copy of Pandolfini's book. Good luck.