First, I endorse thank you letters,notes, emails etc. I’ll take this literally in that when you said “letter” you meant letter. Snail mail gets noticed by me more than the other means, in that they are so rare. Rare = differentiation. As I write this, I have one on my desk in front of me. I’m also assuming from what you said that you are genuinely interested in that company to the degree you’d like another shot(s) at it
Frankly your proposed note comes across to me as very humble, unequal. You demote yourself to interviewee instead of an equal party to a business networking meeting.
If you take the position to yourself & others that you were networking at the very least, and having a business discussion at the most, your note could change to something like
“Frank, Joe, Sally or whatever (you should send your note to every contact) thanks for having me in for a visit, which I enjoyed. I learned a great deal about the company, and having met you I’m even more interested in working with the COMPANY X team. Then Ideally offer idea or business suggestion to drive a point home that you listened, learned and have enough interest and smarts to add value ..e.g. “On reflection have you given consideration to …..? and/or (now being better informed about what this job is all about the best most pointed thing you can do is with blessing from your contact(s), refer the recipient of your letter to someone in your network who can do the job ..e.g. ” I suggest you contact xxxx as I believe you will find them a great candidate. You tailor these comments to fit each person to whom you send a thank you, to personalize the note. Since it’s highly likely they will mention it to each other, you don’t want your effort to come across as a boilerplate ho-hum cut & paste thank you.
I suggest closing by leaving a door open “Please consider me part of your network and keep me in mind as you move forward”
Then there’s follow up. Obviously if you get responses, & get good vibes from them, keep in touch. That makes follow up easy. Again assuming you like what you heard, you like the company, keep researching them keep an eye on what’s happening, and after a month or so..reach out again. a touch base call, another note etc.
In networking-speak if you got as far as an interview..you didn’t lose, you made a good contact(s). And take the long view. You thought the objective of an interview was to get a job. That’s nice, but you’ll manage expectations better if you consider that the objective of an interview, is to arrange a networking meeting and so on until the objective is to receive an offer which leads to a job. If you condense this to interview, job/no job and go away, then you’re most likely to not resurrect your chances, and hence get no job. A thank you is a nice gesture if sincere. But it’s a great differentiator that will gain rememberance and…grow your network

Don–’thanks for having me in for a visit, which I enjoyed. I learned a great deal about the company, and having met you I’m even more interested in working with the COMPANY X team’, this is right on. In the questioner’s version, the focus is on him/herself when it should be on the company, the work and the contribution you plan to make. We had a client recently who came in 2nd for a position. She sent a hand written thank you very close to your suggestion. The person hired lasted only two weeks. Our client was hired immediately.

Another technique is the 90 day follow up, 90 days being the standard probationary period. Of course you don’t say directly that you hope the person didn’t work out. The message is the same as the original thank you: great company, great group, love to work there.

@Chris: Love your 90 day rule. There are times when offers I got for candidates were rejected. The candidate went to work elsewhere and I lost the placement. Knowing the first week or two are critical – it’s the time when a candidate realizes it’s the wrong job and gets very unhappy – I always contact candidates I “lose” to see how it’s going. I’ve moved such people back to my client because I followed up and they wanted an “out.” But here’s the punchline: My clients don’t always know their offer was rejected. They have no idea the candidate went to work elsewhere briefly… because it doesn’t matter. Everyone is excited and motivated to work together. The rest is finesse. ;-)

I think a key point is to focus on the positive from the situation and to acknowledge that perhaps there is somewhere else that may make sense. Something like, “Having discussed with you my background, perhaps my skills could be applied elsewhere within your fabulous organization,” kind of wording where the idea is that you like the company and wonder if there is somewhere else within that company you may fit. You never know if there are other opportunities that just aren’t widely advertised.

Another thought to this is that there are many cases where something like this is done all the time. Look at how many actors will tell about how they auditioned for role A and ended up being role B. Sometimes we end up in a better place because someone saw something and corrected which route we were following.

Practice some “sympathy for the devil” imagine how hard it is to give people bad news. If you can make them feel better about rejecting you, they may want to help you in return. And take the opportunity to ask about any other leads.