Thanks for the responses. I cleaned it today and here's what I ended up doing. Kind of a mix of a bunch of different procedures I've found:
- Used "Fast Orange" thoroughly to clean the whole wrap. I drenched it in this stuff by putting some on my hands and rubbing it in good. This is the stuff I use to clean shafts so I know it doesn't do wood much harm as it's not water based. This took out a TON of dirt and oil from the linen. I did this a few times until the paper power I wiped it clean with was relatively clean.
- Using a hot air gun dried the linen
- Sprayed thoroughly with febreeze to get rid of any remaining odors. I sprayed it enough to soak the entire linen throughout. I let sit for a few seconds and soaked as much out as I could with paper towels.
- Again I dried completely with the hot air gun.
- Sprayed some starch on a paper towel and wiped into the linen completely
- Using a small iron i ironed the linen
- wiped it down hard with paper towels and also used a piece of leather to burnish it a little.
- Finished with a wad of wax paper burnishing the whole wrap.
The end result came out really nice and feels good and smooth to the touch. I don't have a before picture, but the whole wrap was almost black before, and now most of the original white hues are back.