It sounds like the drain in the lean to is a gully. A gully is trapped which basically means the water that you can see in it is because the gully is a u bend like you would have under your sink or in your toilet. This is to prevent smells from the drainage system. It sounds like the drain isn't blocked ( which is why the water level doesn't overflow) but the drain may require cleaning. A decent company would visit you free of charge to provide a quotation to carry out these works. If you are based in Essex, Herts or London I would be happy to do this.
Kinds Regards Iain, Premier Drainage Company Ltd