Replacing NiMH with Li ion
Try to avoid asking "can I replace.." because sure, you can.

The real question is: Is it a good idea?

Unfortunately the answer is a solid NO.

Reasons why:


The NiMh cells are very likely high current cells, meaning they are designed to be able to deliver the high current that a vacuum cleaner motor needs. If you use "standard" Li-Ion cells these will not be suitable for such high currents so even if they do work they will wear out quickly. You can buy high current Li-Ion cells but these are more expensive than the "normal" variant.
You must add a charge controller circuit. The original NiMh cells are very easy to charge if you charge them slowly. Which is the case as your charger only delivers 200mA. When a NiMh cells is full, at 200 mA it will just get a bit warm and that's OK (not ideal, just OK).
Li-Ion cells cannot be charged with a continuous 200 mA, charging MUST stop when the cells are full. If charging does not stop you risk damaging the cells maybe they overheat and could start smoking or catching fire. As you will need some Li-Ion cells in series to make the voltage required, the charger needs some balancing circuit as well. That's complex and expensive.


You're better off replacing the dead cells with new, same model, NiMh cells or just buying a new vacuum cleaner.