As David Vogt points out in a comment, “dran” is short for “daran” - a word that basically means “at” or “on” some location. The location is assumed to be given in context, so “dran” is - as Duden describes - colloquial way of saying “at [this/it]”. The corresponding English preposition will depend on the context of the location/place (“this”) refers to. This is true for many prepositions, where a 1:1 translation either doesn’t exist or would be applied incorrectly; you will need to use context to determine how to translate them.
For example: If I had to translate “Ich denke daran” I’d translate that as “I think about it” but the following phrase from some song lyrics: “an meinem Fahrrad ist alles dran” would mean (lit. “everything is on my bicycle”~ “my bicycle has it all”).
Strangely enough, I can’t replicate the error you specify. When I use Google Translate, it also says “in” when “dran” is at the end. That said, it seems like a bizarre translation.
Without further context:
I personally would translate it as: “My little tea kettle is upstairs at [the door of?] every room.” I read “dran” as being outside and near each room. But I’ve never heard a sentence phrased like that, and I asked a German native speaker and they also never heard a sentence phrased like that. And even for my interpretation, it was pointed out that you’d be more likely to see it phrased as “Ein Teekesselchen steht vor jeder Tür” (“There’s a little tea kettle in front of every door”).
*Edit: the comments below explain the meaning of the sentence in context of the homonym game and clarify how the sentence is meant to be read. (i.e. “The [ceiling] is attached to the top of each room upstairs.”)