I think the advice about getting in as a hardware engineer is solid. At my workplace, there's a ton of need for people working on specialized hardware for DL, and for people working on the software that works with it (optimizing compilers, etc).
If you are looking to break into the software side of DL, the first two thirds of the Deep Learning book [1] contains all the math you need to know to pass the interviews. Then, it's just a matter of getting interviews; I found that I needed professional experience deploying DL/ML to do that. I got that by doing side projects at work. For instance, we had a long standing operations research problem, and I spent some free time at work implementing a RL algorithm to solve it. I didn't get too far, but I was able to talk coherently about the papers involved and about how I planned to conduct the project, which went a long way.