The idea is sound, but this letter is truly surprising to me. Instead of a succinct summary of what goes into the change, we get a long-winded narrative that reads to me as full of excuses. I think a more useful email would be a breakdown of the time actually spent, and a proposal for improvement, something closer to this (obviously sent after the feature has been deployed, and obviously including additional items for documentation, code review, whatever else takes time in your development process):
Summary
- 0.5 hours investigation and planning
- 0.5 hours feature development
- 4.5 hours manually testing
- 0.5 hours deployment
Proposal
- 8 additional hours now adding automated testing
Obviously, depending on your client, they may prefer a more verbose format or need some more explanation, but probably 3-6 sentences at most.