But "ineo" is the verb, where "inire" means "to go in"... Here the sentence has "er" as the verb, not a motion verb (to go, to come, etc.). So it really seems to have a literal translation of "inside in the house" which is redundant.
It's probably some normal construct in danish but maybe some native speaker can confirm it.
As a native speaker of Afrikaans, I also constantly hear "Hy is binne-in die huis" or "Hy is binne in die huis" (He is inside-in the house) around me [almost a literal translation of Han er inde in huset], almost like you would say "Hy is binne-in die bottel" (He is inside-in the bottle.), or "binne-in die water val" (fall inside-in (into) the water), "diep binne-in die grot" (deep inside-in the cave), "binne-in die aar" (inside-in the vein) although -in isn't really that necessary. Aha, I have something from my Afrikaans dictionary - binne-in : on the inside, completely inside, or deep inside. Being "inde in" might mean being inside and being... enclosed, completely surrounded by walls, isolated from outside. Perhaps not easily accessible from outside. The Dutch Van Dale dictionary (1983) agrees: binnenin: "in een afgesloten ruimte" (in(side) an enclosed space).