Air flow is vital to prevent rot and smell. The kitchen areas in 1930s houses were usually concreted but even in those days the sub-floor void of the suspended timber floors was usually ducted through the concrete to an outside wall. Are you sure there are no vents covered over? Or maybe the floor was concreted more recently by somebody that didn't install vents? Either way, a solution would be to cut through the concrete floor and install ducts/vents. It just depends on the layout of kitchen and what disruption that might cause? Ideally you'd have three vents.
The water is a different problem. More houses than you think have excessive water under them which isn't evident because of good ventilation. If you solve the ventilation issue you may find the water issue recedes sufficiently to not be a major problem. If it remains an issue you will need to find some way of creating a drain. There are only really two options - gravity or pump. From your description it sounds like you would struggle to get a gravity drain in? Alternatively you could create a sump with a sump pump discharging to a mains drain. It's not as difficult as it sounds and loads of buildings have them.
The vent issue is an issue regardless of the water so I would tackle that first and see what happens.
