Why don't we reuse spent nuclear fuel?

The reuse of spent fuel involves opening up the (spent) fuel elements and then processing the "insides" to recover unused fuel. The primary reason we don't do this is that it is extremely expensive to set up and run a facility that does this. It is generally cheaper to mine uranium out of the ground, process it to win the metal, and then enrich it to recover high percentage U-235 for use in making nuclear fuel. We might make a comparison to "simple" mining, refining and enrichment by looking at reprocessing.

Reprocessing spent nuclear fuel involves exposure to highly radioactive materials. These pose a number of risks, and this means increased costs. The chemistry of reprocessing translates into costs for materials (like nitric acid to dissolve spent fuel in "step one" of the process). The end of the process leaves a stream of highly radioactive materials that need secure and long-term storage, and this does not come cheap. Though usable fuel has been recovered and reused via reprocessing, it's expensive. Reprocessing fuel is generally not as cheap as a "once through" fuel cycle where spent fuel is just pulled from the reactor after use, left sealed in the fuel elements, and then placed in long-term storage. Unless the price of uranium goes sky high, which, though possible, is unlikely.