When my levers get close to the bars, it is almost invariably because the pads have worn thin (Shimano Ultegra in my case).
I've never had brake pads last anywhere near 2200 km riding; admittedly not at all flat. It costs nothing to pull them out and take a look, and if you have worn them down, you will eventually ruin the rotors, so it is worth keeping track of this.
I think any DOT 5.1 will work; just make sure 5.1 is what is required. The bleed kit will make it easier/possible, but the fluid is corrosive. Wear eye protection and gloves and wipe up any splashes or spills from the bike immediately.
Or just get a bike shop to diagnose the problem in person, and do whatever is needed.