Glad you’re feeling better!
It’s not the most natural construction but is one whose general meaning is understandable. For native American English speakers, at least, the phrase backed up meaning stopped or overflowing (as opposed to computer files being copied elsewhere) is commonly used to describe a toilet being clogged. If your recent illness may have involved any sort of bowel upset, different phrasing is definitely in order.
Your mail piled up because you probably felt too ill to even check the mail, that is, to walk to the box and see whether you had any deliveries. You might express that as
After finally feeling up to checking the mail for the first time in four days, I enjoyed reading the get-well card that everyone signed. Thank you for taking time for your thoughtful gesture.
If you tend to be more private or understated, then
Now that I’m able to catch up on my mail …
Catching up with the mail …
Note that the English word mail is a mass noun or uncountable noun. This means
mails is incorrect. Whether it’s a single letter, parcel, or many pieces of mail, grammatical usage refers to it as a single whole, e.g., “The driver delivered the mail.” The same is true for email, which derives from electronic mail, that may contain notes, letters, memos, and so on. Error on this point ( three emails) is widespread, and correcting it may induce non-responses such as “it’s just different” or “don’t be a prescriptivist.”