As it is only a small area a good flat mopping system with a microfibre head should be sufficient to keep the area clean on a maintenance basis. It may be that the current cleaner is using the wrong type of chemicals and not rinsing the floor properly, which in turn attracts the dirt. If there is scale build up on the floor it will need the appropriate type chemical to disolve it, followed by a rinse. This could be done using a mop and wet & dry vac . You will need to be carefull that the chemicals do not enter the pool itself as it could affect the ph level of the pool. When you say vinyl I presume it is a safety floor which can be quite difficult to clean without using some sort of scrubber but not impossible.