I understand that its not easy for you. I notice, though, that this letter is mostly focused on what you want to get out of it, with a nod in their direction. I can't imagine the circumstance, other than a soft heart, which would make this a call to action to hire you. It always takes time and effort to get new employees up to speed, so this is not a situation of no cost to them.
How would it be to narrow your focus a bit more, work out who is using the systems where you would soon be an asset, and target them, once you know what you could do for them. Then, ask for 15 minutes to talk to them about the issues they are currently facing using and where they see it going.
If you can do this to the people selling / developing the software/applications, you might be able to work out who their customer are. Check out their web site for the grateful thanks from customers section for some leads. You ask these people for a short meeting to discuss rather than your training needs.
You then pitch yourself, either immediately if you can do it or in a follow up letter/meeting, to say what you could do for them. And yes, you have to pitch yourself rather than ask them to train you. Employers will know implicitly that there will be training involved going forward, we want to know that you have the right attitude up front, so that the employer will get the benefit of the training they invest in you. Get your pitch really clear, and work out what you can offer to do for them, to demonstrate your skills and do ask the question, if you are happy with 'this report' or 'that demonstration model' will there be a space for me.
And, yes, you are then going to have to work your socks off, and keep a diary of what has been promised to whom, don't stop when you get the first bite, keep a few irons in the fire til you get a real offer.
Not easy, hope this makes some sense, all the very best.