As most of you know, email is very insecure. Even with a SSL-secured connection between the client and the server that sends an email, the message itself will be in plaintex
If HTML format is used, you can have the message reference assets that you can remove at a later date. If the message is open at a later date, the user should see broken links..
The self-destructing part is really hard, because the user can take a screenshot and store the screenshot unencrypted on his disk, etc. So I think you have no chance to enforce that (there will always be a way, even if you link to an external page). But you can however simply ask the recipient to delete it afterwards.
The encryption is on the other hand is not a problem at all. I wouldn't rely on TLS because even when the sender and the client are using it, there might other mail relies who don't and they might store the message as plain text. So, the best way would be to simple encrypt it explicitly.
For example I am using GnuPG for (nearly) all mails I write, which is based on some asymmetric encryption methods. Here I know that only those I have given explicitly permission can read the mail, and since there are plug-ins available for nearly all popular MUAs, I'ts also quite easy for the recipient to read the mail. (So, nobody has to encrypt the mail manually and might forgot to delete the unencrypted message from the disk...). And it's also possible to revoke the keys, if someone has stolen your private key for example (which is normally encrypted anyway).
In my opinion, GnuPG (or alternatively S/MIME) should be used all the time, because that would also help to make spamming more difficult. But thats probably just one of my silly dreams ;)
If the recipient knows that the message might become unreadable later and they find the message valuable their intention will be to preserve it, so they will try to subvert the protection.
Once someone has seen the message unencrypted - which means in any perceivable form - either as text or as screen image - they can store it somehow and do whatever they want. All the measures with keys and so one only make dealing with the message inconvenient, but don't prevent extracting the text.
One of the ways could be to use self-destructing hardware as in Mission Impossible - the hardware would display the message and then destroy it, but as you can see it is inconvenient as well - the recipient would need to understand the message from viewing it only once which is not always possible.
So given the fact that the recipient might be interested in subverting the protection and the protection can be subverted the whole idea will likely not work as intended but surely will make dealing with messages less convenient.
The problem, as you describe it, does sound very close to the problem addressed by Vanish, and discussed at length in their paper. As you note, their first implementation was found to have a weakness, but it appears to be an implementation weakness rather than a fundamental one, and is therefore probably fixable.
Vanish is also sufficiently well-known that it's an obvious target for attack, which means that weaknesses in it have a good chance of being found, publicised, and fixed.
Your best option, therefore, is probably to wait for Vanish version 2. With security software, rolling your own is almost never a good idea, and getting something from an established academic security group is a lot safer.
If your environment allows for it, you could use the trusted boot environment to ensure that a trusted boot loader has been used to boot a trusted kernel, which could verify that a trusted email client is being used to receive the email before sending it. See remote attestation.
It would be the responsibility of the email client to responsibly delete the email in a timely fashion -- perhaps relying on in-memory store only and requesting memory that cannot be swapped to disk.
Of course, bugs can happen in programs, but this mechanism could ensure there is no intentional pathway towards storing the email.
IMO, the most practical solution for the situation is using Pidgin IM client with Off-the-Record (no-logging) and pidgin-encrypt (end-to-end assymetric-encryption) together. The message will be destroyed as soon as the chat window is closed, and in emergency, you can just unplug the computer to close the chat window.