Soldering fine tungsten wire
I would first look at welding under a microscope using a tweezer handpiece. 



If you don't have access to such a welder with fine low current control, contact the makers and they should be able to help you. The machine should be under $10K for a good one. 

You could also consider using a platinum plated tungsten wire, which would be more resistant to corrosion and easier to deal with. They are used in some kinds of instrumentation.