I don't have any fancy brush cleaning mitt that I use, although I'm thinking about it buying one, my hand works just fine as a cleansing palette. I've been using the Dr Bronner Magic Soap for a few months now, and holy cow this is good stuff. It comes in an array of different scents, so if mint isn't your thing they have a few more to pick from. They come in all different sizes, I have the 237ml which varies in price depending on where you buy it, but it's always under £10, so it's pretty cheap stuff. I used to use a brush cleanser from Laura Mercier which is twice the price and didn't clean my brushes half as well as this.
To start out I just pop about a 5p sized dollop on the palm of my hand - that will vary on the size and dirtiness of the brush - run my brush under some water and swish it around my hand. It's important that when wetting the brush you try not to get the barrel wet, as this will interfere with the glue holding the hairs in place. This isn't a super harsh soap so it's going to leave your bushes really soft once they're dry. I alternate between washing the brush in my hand, then squeezing the excess soap off under some water, and I'll do this until I'm happy they're clean. You can go over certain brushes a second time if you feel it's necessary, but I hardly feel the need when I use this method.
When it comes to drying your brushes, I just run them over a cotton pad or some kitchen roll to take off most the excess water, then lay them on a counter top so the brush head is dangling over the side. If you lay your brush head flat on a surface it's going to disrupt the shape which you don't want.
I hope this was helpful for those of in a bit of a brush rut! It can be a daunting job, but with the right cleanser you can get it done super quickly! I tend to wash mine every 1-2 weeks, well, at least I try to...Another helpful tip is to grab a separate cup, or whatever you use to keep your brushes in, to use for your dirty brushes, that way once you've used a brush and it becomes dirty you know exactly which ones you need to wash when the time comes!