Thankee is a form of thank you, or more specifically thank ye, that is spelt to show pronunciation. In this sense it is the equivalent of the Australian g'day, showing local pronunciation of the more formal good day.
Thankee is encountered in both historical and high fantasy novels and is used to indicate rural or lower social class dialects either in England or the fantasy equivalent.
Merriam-Webster lists alternative spelling as thanky and thankye. The Google Books Corpora show usage in both the UK and US corpora as very small, but with a peak in 1860-1890, and a small increase in the 2000s. My absolutely unqualified and unexpert guess here is that the word became slightly more popular in the latter part of the nineteenth century as a certain sort of 'rural' and 'pastoral' realism for novels became popular and some authors attempted to show English language dialects with some phonetic spelling.
Stephen King fansites list his usage of the word, which might account for both the slight increase in use during the 2000s, and the use of the term on e2 as described by Rancid_Pickle above.