You might get some traction with the landlord if you suggest exactly what needs to be done.

Find the model of the range-hood, and then google "exhaust duct kit" or something related to that model.

The landlord would need to:

- Chuck the extractor kit onto the range hood

- punch a hole in the roof or wall

- run ducting to it from the range hood

- get the stinky cavity professionally cleaned

- put a vent on the roof or outside-wall

As others have mentioned, having range-hoods just vent straight back into the kitchen is kinda normal now, which sucks. So the landlord probably isn't legally obligated to do this. But if you suggest it from the perspective of "this will eliminate the smell and stop your cavity from rotting" the landlord might agree to spend the money. The landlord probably does have some legal obligation to get rid of mould and stench.