A KIL operation stops the processor because the internal PLA never signals an end-of-instruction. Thus, the processor doesn't actually stop, it just enters an endless loop waiting for a non-existant end-of-instruction signal. In theory, therefore, the instruction takes an infinite number of cycles.
After the 1st 7 cycles of a KIL instruction, the internal PLA will not activate any signals so at this point the processor is doing nothing. What happens in the first 7 cycles depends on the internal structure of the 6502 PLA. It is possible that internal registers are updated during these cycles.
If you wish to find out exactly what a 6502 does in a KIL instruction the following link may be helpfull
http://www.pagetable.com/?p=39
It is a discussion on how the 6502 handles undocumented op-codes.