Dear Lilly's Mom,
I'm so sorry about the physical loss of your precious Lilly! There is nothing like this kind of grief. And I'm sorry your other Dachshund is grieving. I'm sure you 2 are a huge comfort to each other! Do you have other people or pets in your home as well?
Please know that Lilly is still with you, and she is free of any pain. She's a free little spirit--with you and anywhere else at the same time, because in her realm, there are no time/space limitations.

She feels your love for her.
When it is your time to pass, you'll be fully reunited. In the meantime, she wants you to be okay.
I hope you keep in touch here.
Thinking of you. Prayers of peace coming your way,
Kathy