Check your battery's health/age, then check to make sure the connections are clean, greased & tight.
Cut a little bit of the insulation on the large red wire going to the starter.
If you see white corrosion, you should plan on replacing it - possibly the negative cable, too.
Own a VOM and check the battery's voltage before charging, after charging, and most importantly, during starting.
If all of your connections are clean & tight (don't forget the other end of the large positive cable, under the starter cover) and the starter turns over, you will get an idea of your battery's health.
For grease, use dielectric grease (auto parts stores), axle grease, or even vaseline. It is only to repel moisture.
I like your K75! I have never ridden one that I can remember, but they are my favorite of the water-cooled BMW's. I think you will find the R65 to be a worthy stablemate if you have the room! With fresh tires and a good-running engine, it is quite a lot of fun! Especially with the lower European bars (you have the handlebar BMW made for the USA and old folk...). The swap with OEM parts is not cheap (used is generally preferred), but it is easy and can be reversed just as easily.
The result is substantial, but still comfortable. But I guarantee you you will be wanting rearsets! lol!
This picture shows the stance provided by the BMW Euro bar:

There are cheaper alternatives to BMW's bar, though.
However, it is important that the bar be 22mm in diameter for the control perches and the clamp at the triple tree. Most Japanese handlebars are 7/8" (why???), and of course - H-D are 1", just because bigger is better in 'merica.
This is a company in the USA that offers bars for all three sizes:
http://www.sideroadcycles.com/ImportedMotorcycles/ImportHandlebars/ImportHandlebarsBMW.htmlExpensive - but a good reference guide. I'm sure you have an idea for sources in your area?
Welcome - and congratulations on your new purchase!