Well...
I don't even know if there are 10-speed downtube shifters available.
But if there are, I still don't think they're such a hot option for you. 9-speed will give you a lot better bang-for-the-buck and be a lot more available too.

Either way that crankset has to go. It's explicitly stated as needing an 1/8" chain, which isn't compatible with any multi speed cassette.

Not knowing what your friends intends to ride, it gets a bit hard to advise. We don't know how much money you're willing to sink into the project either.
One option would be to go for an Alfine IGH rear wheel. It could be made to play nice with your crank, and you might even be able to use a bar end Alfine shifter and stick on the downtube.
Or pick up a dirt cheap hybrid from Halfords or something, transplant the driveline, then try to source a DT shifter with the correct speeds from Ebay.