Replace the IR led with either a logic-level small-signal mosfet or an optocoupler. Hack one of those headphone cables to get access to the switch and rewire it to mosfet/optocoupler. Replace the bit in the code where the IR signal is sent with a simple pulse given to the mosfet/optocoupler.
That's the gist of it. Schematics are available if you search this forum a bit, because the question "how do I replace a button on device X with an Arduino" pops up once or twice every week here. Component choice is also discussed here and there.