Replying to a thank-you letter is not required by Western etiquette, though letting the sender know that their letter arrived is a kind gesture. If you receive a thank-you letter and want to reply, you can do so in whatever form you wish: by letter, in person or electronically.
According to Miss Manners, it is not necessary to thank someone for sending you thanks. However, if their thank-you letter was accompanied by a gift, such as flowers, chocolates or photos, then you should write a letter of thanks in response to the gift. If the thank-you letter contained something other than just thanks, such as news or updates about the sender's life, you can reply to the news however you feel most comfortable: in person, by letter or electronically.
Etiquette guru Emily Post says that thank-you letters should be sent to people who have hosted you in their homes, given you gifts, or sent written condolences after a death. In general, thank-you letters should be promptly written by the person doing the thanking. However, If the person who received the gifts or condolences is unwell due to bereavement or illness, a close friend or family member may write the thank-you letter on their behalf.