All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The scholars stated that it is neither permissible to participate with the non-Muslims at any of their festivals nor to congratulate them on this occasion or to show any kind of happiness for it. Therefore, if this Rakhi is among the festivals of the Hindu, then your sister is wrong in writing this expression for you because doing so is confirming the falsehood they are upon. Why is it that she is missing you on this particular day if the intention was not to glorify this day? Therefore, what you mentioned is absolutely evil and she must repent for it.
For more benefit, please refer to Fataawa 93861 and 83628.
Nonetheless, you should be wise and lenient when blaming her for this act, because whoever commits evil out of ignorance, he/she should be taught with leniency so that this would not make him/her refuse advice or guidance.
Allaah Knows best.