How do I prevent water from collecting against side of house?
If the cost of raising the patio is too great you could get an angle grinder or perhaps a better choice would be renting a concrete wet saw and cut out a strip about 6 inches wide from the edge that meets the house wall. As @woodchips points out in his comment, make sure you wear gloves, safety goggle and a mask for this. It'll get hot, but it's very dusty work and the saw will throw up chips of concrete.

Remove all the concrete and put in a gutter that will collect the water and direct it away from the house to the drains.