I second the idea of taking a day out to blitz the house. Once it's clean and organised it's easy to keep clean.

Personally, I'm a neat freak and have to keep things tidy, other wise I feel out of control (actually I could be a little OCD about it all, or at least DH thinks I am LOL). But what works for me is doing everything as I go. Get out of bed, make the bed. Make breakfast, clean up breakfast. Get a load of washing out of the dryer, fold the washing. Put DS to bed, clean up the toys etc. It means that you do have quite a few 5 minute tasks here and there, but that's all it is, 5 mins at a time. You don't get to the end of the day and have a huge list of housework to do.

Then, I would take your DP's advice and draw up a cleaning roster. But, I would make sure that the tasks are evenly shared between the TWO of you. Maybe even have rotating weeks so one of you isn't always stuck cleaning the bathroom.

HTH