so...really...all we need to do is figure out which ion of Li+, Na+, K+, Rb+ or Cs+ is in the solution.
I have two suggestions, and if someone could comment on each's applicability, that'd be awesome:
1. First mass an empty beaker (accurately). Put the solution under a heat lamp, to cause slow evaporation. Measure the volume of solution you have when precipitate JUST starts to form (if this is even possible - the precipitate would have to be macroscopic i.e. visible to the naked eye). Then cause slow evaporation of the REST until just the crystalline form is left, from which you can get the mass of crystal which was dissolved. Use known Ksps to determine which salt it was.
2. MAYBE we could just use the common ion effect. If, say, the dissolved salt was LiCl, then LiNO3 won't be nearly as soluble in it as it would pure water. So if you divided the solution into 5 portions, and tried adding the soluble nitrate of each alkali metal to one of the samples...ONE will precipitate earlier than the rest - the one which already had ions of that type in the solution.