How likely does this work:
Accelerate neutrons to as high energy as possible (particle accelerators/ cosmic rays ...) and hit (small samples of) our radioactive waste with it such that it is converted to non-radioactive elements (possibly while winning energy).
I found that the principle is known as accelerator-driven system (ADS).
The application of active deconstruction schemes to manage nuclear waste has long history as a proposal, but has not yet been demonstrated as technologically practical. The usual schemes (i.e. the ones people have put time and money into and are still looking at) use electron beams (either directly or as a bremstrahlung source) because electron beams are easy and relatively efficient to make.
The underlying technological problem with your idea is the line
Accelerate neutrons to as high energy as possible [...]
because there is no simple way to accelerate neutrons. Our accelerators work on charged particles. Neutrons beams have to be generated as a side effect of some other process and that adds a completely unnecessary inefficiency to a process already of marginal efficiency.
The transmutation of nuclear waste is a well developed research area. There was extensive work done a Dubna regarding this area.
If you're looking to get into this area, there is the publication "Nuclear Methods for Transmutation of Nuclear Waste: Problems, Perspectives, Cooperative Research" published back in 1996.
