If we actually funded more research we could probably solve that problem. They have reactors now that can make use of recycled waste. Improve that technology. Also we could launch that stuff into space in blast proof containers.
First, little progress has been made since I heard a talk on the matter in Toronto at the GAC/MAC/GSA convention Short Course on Uranium Mineralogy. We really don't have a solution that doesn't have problems. There have been no real new ideas since then, mainly because the geology of the planet just hasn't changed that much since '78.
As for shooting high level nuclear waste into space, not only is it heavy, but one launch failure would make a mess. Unlikely? One in 50 shuttle missions ended in disaster. (Challenger, Columbia), so shooting our messes into the great beyond is a non-starter.
Keep in mind every atom you split produces at least two more unstable (radioactive) atoms.