First, although her name had the variant form Jana, the Roman name Dianus was relatively late, whereas it should be early if it is to link with her. Second, he is a native Roman god, whereas she was an imported goddess (from the Greek Artemis). Third, Diana seems to have no connection with doors or two faces.
It could well be, however, that doors and gates came to be associated with Janus (rather than the other way round) because of the similarity between ianua and ianus and his name - which in this case will need another explanation. We do at least know that the month of January was dedicated to him, not because it was 'two-faced', looking back at the old year and forward at the new, but because it fell between winter and spring.
