Well I think it was near the end because she thought that it was ridculous about Leopold being in the year 1896 but when she saw the photo that Charlie gave her she realised that he was in fact telling the truth about his 19th century life. Also Charlie was the first one to spot it as he told Stuart wait isn't that Kate....and when Stuart saw it he was like Damn it ........

I think she began to believe then, when he caught the thief, she realized Leopold's a sort of knight, gentleman so he was that man who isn't ordinary man in 21st century..
but when they had dinner at the roof they were talking about his life so she made sure of that and only when they spent that day together and he showed her his things from 19th century etc. I think she was absolutely sure that he's from 19th century when he showed his home..
Although it's difficult to say what moment particularly it was..

Honestly, I think it was when the elevators started working again.

Oh, maybe, sorry that there's no such variant..

But it's a difficult question indeed... She might have started believing that moment, but it could be also when she saw the photos...

I think she was still a bit skeptical in this part, but it was when she first started to have doubts about her opinion that he was just acting.

I think so because she said something about jumping, and that is a leap of faith.

Like I have mentioned before in my old profile. She thought that Stuart(her ex) was looney and she had no idea that Leopold was who he said he was a 1876 duke from the 1800s.