Typing 110-120 wpm, your fingers are hitting ~10 characters per second which is a pretty impressive feat if you think about it; most people will never be able to achieve much beyond that, even with dedication & practice.
One thing you can do is ensure you have a decent keyboard that you're comfortable & familiar with. I'd recommend a mechanical keyboard with Cherry MX brown switches (light tactile feedback without the annoying 'click' of blue switches) - but ultimately, keyboards are highly subjective.
To each their own, but tactile keyboards are usually favored by typists because there's physical feedback from actuation. Also, MX browns are actually strictly speaking a tactile switch, just the bump is very muted.
The fastest typists these days swear by Colemak-DH which just like Workman fixes many of the shortcomings of Colemak.
To me Workman feels the best, but anything that's not QWERTY is measurably better, whereas the difference between Colemak, Dvorak and so on is more circumstantial.
No need for labeled keys, it's all about developing muscle memory. Just practice touch typing, starting with the keys near the standard position. Your typing speed will naturally increase over time as your brain adapts, so don't get hang up on not improving fast enough on any given day, sometimes you just have to sleep on it.
So I would say it takes one month... To learn. . It doesn't matter to have the correct label on keys... You are not supposed to look at the keyboard anyway.
Do you use Dvorak on mobile as well? Is it a pain to use QWERTY devices when you encounter them?
This kind of learning is procedural memory, it's like riding a bicycle, you don't forget it, you just get a bit rusty over time. You will be able to use both layouts.
There's a 2-3 second timeout to motivate you to type fast. Also, it pulls words from the buffer you have open, so you can practice on a set of words that you use regularly. It starts with short words and gradually moves to longer and more complex words to see if you can keep up.
Of course, it will only teach you Dvorak, but then again, you did say that you wanted to type faster...
I don't think my Dvorak is faster than my QWERTY. (About 100 vs 110 wpm on the casual typing speed test websites).
But Dvorak is absolutely more comfortable.
Wearing some long-sleeved thing will make a difference, but that could make you hot; sports sleeves are a way of adding a layer just to your arms without making you hotter anywhere else.
It's fine to be typing slower during practice. It's like how sports athletes will practice a pass or shot in artificial conditions, rather than play a full game.
In terms of using real words: I don't mind that I'm typing slower per se, but I do notice when I do typing tests that involve real words, I have a certain muscle memory (at least in the common parlance — I assume it's actually more brain training) that allows me to type many long words quite quickly.
I assumed that it would be useful to improve my technique using real words, with a focus on the words that I'm not great at. I will be interested to see if typing gibberish words with similar patterns will lead to improvements in real world typing.
The discordant nature of random words just doesn't flow well and you end up stumbling (mentally) because of it.