I'm sure there's something in consumer law about this sort of thing.
In the first instance the servive provider should be given the opportunity to put right what is wrong, before getting someone else in.
The problem here is that I see a right barmy breaking out between 4 parties. The first carpet cleaner says it's a good job, the tennant says it's a good job, the landlord says it a bad job and then they ask you

Which ever way you go you're going to be upsetting someone.
If you do decide to go ahead with it, you'll want payment off the landlord complete with satisfaction signiture and if you think it (previous effort) was a bad job then get some good photos done to show the areas of concern both before and after, because situations like this often end up in court.