I have just popped open a 5 Cell 18v Li-Ion battery I have here on my desk (From a TITAN 18W31), working on the assumption that there is very little variance in the design of these things.
Looking from above, with the open end of the battery contacts facing away, I have from left to right B+ | T | BS | B-
B+ is the +18v feed from the back, B- is the "ground" at 0v
T looks to be (and guessing from it being called T) a data/reference voltage used by the charger to balance the pack (it has an onboard cell balancing IC). This one is not connected to the drill anyway.
BS (Presumably Battery Sense) is connected to the drill (although mine operates with it disconnected also). It comes in to the battery and immediately into a temperature probe wedged between the cells. The other side of that probe is connected directly to B- ("Ground"). When the drill fires up, it sends 5v back down that line into the battery, and by extension through the thermistor to ground. The thermistor (at my current room temperature measures 11kohms. Its resistance reduces at it heats up. This explains why my drill still works without it, it is looking for a resistance to "Pull" the 5v it sends to the battery down toward ground as the battery gets dangerously hot. Since not being connected doesn't pull the value down, it still works.
My suggestion to you would be,
1) Connect power to your switch. Check if some voltage appears on the mystery pin.
2) Buy or steal a 0-10k or 0-100kohm variable potentiometer, set it to 11k (using a multi-meter) then connect it between the mystery pin and ground. Twiddle it, see if it fires up.
3) you could also use a "variety pack" of fixed resistors from ebay, but it involves more twisting stuff together.
Lastly, power electronics for Lithium batteries are made to handle quite a bit of power. That means if you connect stuff wrong, the resultant bang is quite big. The above is a guess only, make sure you are confident and competent when dealing with these things, because I am just some guy on the internet - and worst case a decent short could hurt you or start a fire.