Unless you are a real train buff, and passionate about this, consider this:

A Nozomi train will take 2hours and 20 minutes (27,000yen return trip)

A Hikari train will take 2hours and 37 minutes (and cost about 600yen less)

The best advice would depend on whether you are able to change your inbound or outbound flight to KIX.

If you can't, most sensible and economical course of action would be to grab a two or three day deal from JTB Sunrise (you can book through IACE, I've used them myself and they allow you to book with a foreign credit card - if you're in the USA, they have offices there).
I've spoken to IACE about extending one of these trips so that you use the hotel for the one or two nights and go back by train on a date of your choice and they do allow this.

As this deal limits you a bit as to which trains you can use, if money is no object and you'd prefer to dictate your own timetable then you can just buy your ticket when you arrive in Japan for the Nozomi of your choice. There are plenty of hotels in Kyoto to choose from, if you want some help, just let us know the price range you're looking for and if you'd prefer Western or (generally more expensive) Ryokan/traditional accommodation.

Please jump in someone if any of these dates coincides with a national holiday that Mr. & Mrs. N.U. should be aware of!