There is some variety of opinion on what is required for a conversion to be halachically valid (example of such dispute) - rather than take an opinion based on who I feel should be counted as a valid convert, I'd like to read up on the relevant sources on the topic. What are the most relevant Talmud and other passages which inform thinking on this topic?
I advise you to begin by reading the Tur and Bet Yosef Yore Dea 268-269 . In Bet Yosef you would find most references to Talmud and further you can learn deeper in the Talmud itself and extend the study to Rishonim. In a third time learning Rambam following the annotations En Mishpat. on Talmud, and to re-read Tur Bet Yosef and Shulchan Aruch with Beur Hagara and Shach. If you have a couple of precise questions perhaps I can help you more.