I also had the exact same issue recently and the advice I got from a member which is below helped me fix my ignition switch. Expect to pay about $200 for it. Surprisingly the part number is still available from BMW NA and I purchased mine from my local BMW dealer and they had it to me in two days.
The piece that has the soldered wires to it, is crimped on to the switch cylinder that is likely still inside your steering column. That assembly is held on by a very tiny grub or set screw. Jewelers sized flat head. A picture of the switch is depicted in the diagram below. No 5 is the grub screw and 7 is the switch assembly ( 61321355148 )The switch is available, but can be pricey. The switch is separate from the key and lock assembly, meaning the only thing that you might have to replace is the switch. Per the additional picture, across from the purple wire is the hole that receives the grub screw. Look carefully.
You still may be able to salvage things. You indicate you have an 02. The switch removal is much the same in the two models, so you might want to look at the 02. The switch design is similar too. (Wire lengths and end connections are different).
Before doing anything else, compose yourself, and with some decent lighting and maybe even a magnet, look to where the parts came apart and may have landed. (If the grease was gooey enough, the grease may have retained a few of those parts in the steering column.) From memory, there are two angled or bent metal tabs that fit into the brown/black bakelite and perhaps two or three springs (similar to those found on ballpoint pens). I think you are looking for 4 or 5 parts in all. If you locate these things, it is a very simple operation to press fit everything back together.
Good luck.