Good stuff!
Are you planning to use the mouse power supply, or the button cell power supply?
If I'm trying to save time while modifying a larger number of mouse quickly, this is what I prefer doing:
The quick more idiot-proof more accident-proof (button cell) technique is simply find two contacts that correspond to the mouse button. Once the multimeter leads are attached [solder bump, or button connector, or ribbon connector pin, etc] I then press (or short) the mouse button switch and watch the multimeter screen.
If using the coin battery plan, I don't even need to plug in the mouse (yet), or attach LED circuit (yet), to test for where I need to attach two wires. This is what makes the coin-battery technique so safe, easy, idiot-proof accident-proof when teaching an electronics newbie how to modify their mouse.
Once I find the two button attachments (usually at solder bumps underneath the button or pin connector) where voltmeter resistance that goes (near) 0 ohm when button pressed, (near) infinite-ohm when button released, that's where I attach two wires. Then reassemble the mouse.
That said, I usually plug in the disassembled mouse briefly anyway, to make sure that the switch I pressed, is actually the correct mouse button, and to determine the voltage polarities. And yes, to test LED before reassembling. But it is not necessarily mandatory to make the mouse live again until the wires are fully soldered and mouse reassembled, if you're very paranoid about plugging in a disassembled mouse. Voltage polarity can be tested after you finish modifying and reassembling the mouse, and safely attaching multimeter leads (in DC voltmeter mode) to the two wires coming out of an assembled mouse. This allows you to find out which polarity to safely attach the external button-cell-powered LED. (This avoids a lot of newbie mistakes: Accidental slips of blunt voltmeter leads on a cramped mouse motherboard, while the disassembled mouse is plugged into the computer, accidental shorts are common newbie-mistake when working with live electronics circuits). Even in the event of polarity mistake, the existence of LED -and- a resistor (yet another good reason to have a resistor, "just in case"), will keep accidental reverse polarity current low enough to not damage electronics. But you won't do that, because you'll multimeter-test a live reassembled mouse before attaching the LED circuit.
That said, experienced electronics engineers will definitely strive to avoid the button cell, and do more testing/analysis necessary to fully bypass the need for the button cell.
(Literally 10x as much multimeter testing, depending on how complex the mouse is = lots more work when you're trying to mass-modify a large number of random discarded mice. When this is the goal, it is much faster and short-circuit safer, because of far less multimeter testing needed especially for clumsy hands)
Using the mouse's power supply to illuminate the LED, please keep corresponding here, so we can make sure you modify your circuit safely (good stuff, Sparky) -- Just adding further commentary to explain why the button-cell technique is useful for reduced risk/newbies/idiot-proofing/rapid random assembly purposes... Once done, take pictures of your final resulting circuit, so others will be able to take advantage of it for the same model mice.