I'll start by saying, you must have long arms.
Didn't you want to clutter the dashboard with a horrible aftermarket switch?
If you bought a 3 prong terminal illuminated switch then you've probably bought the correct one. (You can also buy 3 terminal switches that are single pole/2 way which you wouldn't want).
These 3 terminal switches aren't designed to illuminate when the dashboard is lit up at night(4 terminal one) so I wouldn't connect it up to the ashtray illumination wire as in theory it might work but you could easily overload it and blow the fuse . (I'm not sure of the fuse size).
Did you identify the terminals on the new switch, it looks to me that you may got them mixed up and have blown the fuse for the ashtray?
The 3 terminals on the switch should be labelled:
1.Earth............. Connect it to a good bare metal earth somewhere around the centre console.
2.Load...............Connect this to the foglight. (The other connection at the foglight itself should be earthed).
3.Supply............
Connect this to a 12v supply that becomes live when the ignition is on (This is obviously to prevent the battery from being drained if you connected it to a permanent live and forget to switch the foglights off), also fit an inline fuse. Don't forget you're adding an extra load to an existing circuit that supplies something else so make sure it's man enough. (The ashtray illumination isn't).
I can't remember which 12v supply I used last time as it's a while since i worked on the electrics, I vaguely remember using a supply in the drivers footwell which was there for heated seats which I don't have.