The injector is a DIY and runs $150 or so IIRC. Make sure you get the coupling element and find a local coder to help you code it into the DME (or do it yourself in INPA).
IMO if you have a leaky injector your car would stutter and idle poorly on a cold start and throw misfire codes. I am thinking valve cover gasket (this may be a $400 dollar job on labor alone).
-edit white smoke pouring out of the tailpipe in this cold weather isn't necessarily anything to worry about. Likely just condensation build-up in your mufflers. If your fan goes full blast and you get a red check engine light, then your water pump took a dump. That's not a cheap fix ($800-900 including labor).
E90 LCI N54 6AT