Ok, I'm willing to give that a shot.
What do I need to connect? Do I connect the programmer's Vcc (5v), MOSI, Reset, SCK, MISO, and Gnd to the respective pins on the ATTINY44? You mentioned 220 ohm resistors, where whould I put them? I see plenty of examples online how to program a chip in isolation, but I can't find anything about programming a chip that's soldered into a production board. I assumed that's what was meant by In-System Programming, but I can't find any good examples.