I am not an expert but my guess is that in the Akihabara Electric town there aren't any. But the greater Akihabara area might have some. Though, keep in mind that it is one of the busy areas of Tokyo city and even if you find an Akihabara onsen, it will not be the same quiet experience that you find in the mountains :)

There is one Hotel Dormy, halfway between Akihabara and Okachimachi stations, with an onset on the rooftop. But this one is not natural. As @BanzaiTokyo mentions, this will not be as calm and relaxing as one in the mountains. For that, you can visit Mount Takao or Mount Hinode, about an hour from Tokyo.

I used to love an nice memorable old-style kuro-buto near Ueno station, that I found by chance a few years ago. Now that I don't live in Tokyo anymore I don't remember exactly the location ...

Few searches on the web and I found this one, not so far from Akihabara district: Tsubame-yu, between okachimachi and ueno (3-14-5 Ueno, Taito). Looks classic and simple, but has even a website, check it out!

Thanks for sharing! I am curious, what's the link?