Are your reports, utilization, and scores in order as such that a graduation is appropriate. You’d hate to have them review your report only to see high utilization and not much score movement, for example.
^^^This is the most important part.
So, if you havent sent email yet, as others said, redo so it's concise and to the point.
And kudos to you for getting back on track with your credit profile.
I don't think there's anything wrong with what you wrote, per se, but there is a lot in it that a bank wouldn't be concerned with. The three things that matter to them are profit, cost, and risk. And maybe customer loyalty/satisfaction (depending on the bank). You simply want to show that the risk is lower than it appeared when you opened a secured card. I would trim it to something like this: "I am writing to request that you convert my secured card to the unsecured version. Since opening the card, I have made every payment on time for X months, usually paying my card off in full each month [if true]. My balance has never been higher than X. During this same time, my family's income has risen from X to Y. I appreciate your faith in issuing the secured card to me last year [or whenever]. I value my relationship with your company and hope to maintain and extend it for many years in the future. Sincerely, [your name]."
JMO. Either way, good luck!