There are a few different ways of interpreting your question. Interpretation #1 is "among all characters who appeared in anime from Jan 2015 to Mar 2016, which of them is most popular on the internet by metrics X, Y, and Z". Interpretation #2 is "among all characters who appeared in anime from Jan 2015 to Mar 2016, which of them is most popular on the internet as a consequence of their appearing in the anime by metrics X, Y, and Z". Interpretation #2 is considerably more difficult to measure, but it may be more relevant to you.
It also matters whether by "the internet" you mean "all of the internet" or "the English-speaking internet", because Japanese viewers tend to have systematically different tastes than non-Japanese viewers.
#1 / all
Under interpretation #1 and "all of the internet", the answer almost surely has to be someone from Kantai Collection, which is the biggest thing since sliced Touhou. Take your pick from Shimakaze (an early character who achieved high popularity and has maintained it since); Fubuki (brought to prominence by the anime); Kashima (the newest high-popularity character from the franchise); or any number of other shipgirls. (I think Shimakaze wins by most metrics, including pixiv illustrations - 24000 as of now; this pixiv poll from 2014 by a large margin; etc. But she did have a considerable head start, being ship #10.)
#1 / English
Under interpretation #1 and "the English-speaking internet", sites like MAL are good places to look.
- On MAL, the top-ranking female characters from anime that aired during the time period in question are Senjougahara Hitagi (Monogatari), Suzumiya Haruhi (via Nagato Yuki-chan), Saber (Fate), and Erza Scarlet (Fairy Tail).
- On Anime-Planet (which I think skews to a more casual demographic than MAL), the list is Erza Scarlet, Hyuuga Hinata (Naruto), Suzumiya Haruhi, and Lucy Heartfilia (Fairy Tail).
- AniDB is a bit more difficult to work with since they have ratings rather than just "likes", but the toppers by raw rating (without adjusting for confidence based on number of ratings) are Miyauchi Renge (Non Non Biyori, twice!), Kongou (KanColle), Cure Beat (PreCure), and Cure Moonlight (PreCure)
#2 / all
Under interpretation #2 and "all of the internet", the answer may not be someone from KanColle, because the anime, while still very popular, was not quite as ridiculously popular as the game on which it is based.
Idol shows are another common source of super-popular female characters. Within the time period under consideration, Love Live! and Idolmaster both aired, and both of these shows derive more of their popularity from TV than KanColle does. I know that Love Live! has (had) periodic character polls, but I can't find many of them at the moment. Here's one from Jan 2016 ish which has Nico at the top (see, Japan has weird taste), and another one from back in 2013 that has Maki at the top (okay, not that weird). No idea whether Idolmaster does them too.
Other than that, I don't have any good data, but my impression is that the various female leads from Saekano, Fate, Oregairu, and Monogatari are also potential contenders under this interpretation.
#2 / English
For the English-speaking internet, interpretations #1 and #2 are actualyl mostly the same, since most of the English-speaking internet's exposure to "otaku culture" is through anime, with virtually no non-anime media achieving prominence anywhere near the most popular anime. (Contrast Japan, where KanColle, the game, is very much a thing.)
Perhaps the key exceptions are a select few visual novels like Fate/stay night and Muv-Luv; and wildly popular manga like One Punch Man and Shingeki no Kyojin (though these things pretty much always get anime adaptations eventually). But I'm not aware of any non-anime properties like this during the time period under consideration, so I expect no real difference between interpretations #1 and #2.
One other well-known source for this kind of thing is Newtype magazine's monthly character polls. But I warn you in advance, their polls skew in very strange ways. For example, Kira Yamato is almost perpetually ranked, despite not having been on TV since 2005. Anyway, I found most of these for the time period in question, and it turns out that Saber tops about half the polls and places in the top 3 in the rest.
My personal take
If you choose to include KanColle, I think the obvious answer is Shimakaze, or possibly Kongou, given that you want a character with high recognizability in the vein of Haruhi/Asuna/Mikasa/etc.
If you choose to exclude KanColle, which may also be reasonable depending on your target audience, I would recommend a character from Monogatari, which has had 7 years to stew in the minds of people, yet is still putting out new animated content. Senjougahara is probably the most recognizable of them, but Shinobu/Kiss-Shot is more topical, given that Kizumonogatari is literally happening this year. The other reasonable option I see is Tatsumaki from One Punch Man, too, given the broad popularity of that show with English-speaking audiences.
Alternatively, you could just use Mikasa. Jiyuu no Tsubasa (recap movie #2) did premiere in June 2015, after all.