I started TI drills a little over a month ago now, and I've mainly focused on balance drills, particularly skating, since then. I've made very noticeable improvements in my full stroke in that time, which is in stark contrast to the slow improvements I had made in the year prior to this.
I don't really understand how the skating drill impacts on my full stroke; it's not something I'm consciously aware of, but I can't think of anything else to attribute my development to.
I recommend persevering with it. My advice is to get in the slowest lane, and don't worry about the kick too much -- if you can maintain a good position at a slow speed, then you're mastering the balance. For me, learning to turn my head for a breath while maintaining a skating position was a breakthrough.
Hope that helps.