Advice needed. Our neighbor’s gutters are flooding the side of our house

Hey Houston!

I'm needing your advice. We recently acquired a house near the galleria and we are in the process of getting it renovated. We noticed that our neighbor's house is completely flooding the right side walk way to our backyard. There are two things causing this. First, the gutters on top of their house have a very obvious break where water is literally waterfalling into the side of our house ( I have videos of this). Second, they have another gutter that is just emptying into the side of our house (no drain for the water). This causes a huge pool of water to accumulate by our side gate to the point that it is unusable during rain.

We very politely told our neighbor these issues and we were trying to work with them to install drainage that would take care of this problem. After giving us the run around for a couple of days, I had the unfortunate chance to talk to the neighbor's wife this morning (she wears the pants in the relationship) and she was extremely rude and condescending about my request. She told me the drainage issue was not her problem and she "cannot control where the water falls." Essentially telling me to go f**K myself. What are my next steps? Perhaps the biggest challenge is that our neighborhood does not have an HOA. Do I need to involve a lawyer, etc? Any help would be greatly appreciated