I think Fionwe's explanation below is the best, and is a representation of what I think about it. There is no charity about their living together - there is a deep malignancy present that one would expect to find in the Forsaken.

Aran'gar didn't want to be friends with Graendal. She was irritated that Demandred, Semirhage and Mesaana were in an alliance and she looked towards Graendal as a source of information and possible collusion. She did, however, start to plan Graendal's downfall from the very beginning. It's completely likely that Graendal did the same - even Aran'gar said she'd flirt with someone she planned to murder than evening.

Graendal and Aran'gar are together in the palace because they're using each other. Graendal could detect threats with saidar, and we saw how Aran'gar sensing Rand helped her. It is also likely that together they would be able to destroy most threats. However, they were both constantly planning one another's demise, and Graendal took advantage of it eventually.

Aran'gar knew that if she could kill Graendal she'd get all her assets too, and increase her chance of being Nae'blis. Graendal probably thought Aran'gar good enough for a pet, as a potential but temporary ally, and as a source of information especially considering one of Balthamel's fortes was espionage.