Sure SM57

Its a great mic for snare & toms. Could also be used for hi-hat, overheads and probably even the kick at a push.

A better choice for the kick drum might be a Sure beta 52.

Overheads and hi-hats would probably benefit from condensor mics... would need some idea of your budget to make any suggestions.

The Sennheiser e604 might also be worth considering for toms and possibly also snare: these can clip onto the drum shells and will save you some money on mic stands plus some time setting them up... check with the drummer first though: some drummers don't like clipping mics to their shells for fear of damping resonances, and some drums with non-standard rims won't allow them to clip on easily. (The mics can also be used on stands if that turns out to be a problem) I recently picked up a pack of 3 for £200, which makes them about the same price as SM57s.

I've had great luck with an Audix i5 on the snare top and an SM57 on the bottom.

I've always like the Audix D6 on kick, but I've never recorded a jazz kit before, so I don't know if that would work well for you.

You can't go wrong with theAudix i5 for the snare. For kick drum I like either the AKG-D112 or Shure Beta 91. For toms there's many choices SM57's are always great, Sennheiser MD 421's work great on toms and Audio Technica's ATM350 are really nice. The ATM350's are a clip-on condenser mic and have a very small profile I use these alot on toms and drummers tend to love them because they are very unobtrusive and being a condenser mic they pick up the stick attack very well as well as the resonance of the shell, which gives a good rounded/full sound to the toms. The only way to go for hi-hats and overheads in my opinion is to use small diaphragm condensers such as Audio Technica 3031's... there are many others of course and all these recommendations are of course subject to the limitations of your budget.