The use of ‘last’ in this example is not a natural fit in this sentence. When considering dates and when things happened, the reader will always refer to the date of the letter.
Writing dates naturally in English
If the date which includes the sentence above is February 2018, ‘on’ is more natural than ‘last’.
If the date on the letter is February 2019, ‘on 18th January last year’ is natural or even ‘in January last year’ as the exact date of admission is no longer important.
Make sure you use natural English for the environment where you will be living and working.