From the quoted definitions at etymonline, I would suspect that you may be asking the wrong question :)
If I look at the related words in other languages (dag, Tag) for day, it seems the final g has changed into a [j]. The same seems to have happened with (Dutch) leggen -> English lay.
As it is normally pronunciation that defines spelling, and not the other way around, it seems that the i did the job of representing that [j] in daily quite well. However, as a final vowel, the i seems to be uncommon in English, and usually written as a y. (Compare Dutch hooi to English hay).
So where dæglic became daily (the final [k] became [i]), dæg became day. In both cases the g got to be pronounced as [j], but instead of dai, hai or gai, the use of y was preferred to write that sound in a final position.
I think that in practice, most people would simply use the verb 'scrutinize' instead of saying 'watch with scrutiny', which accordingly doesn't sound very idiomatic to me.
That being said, 'scrutinizingly' does exist; it means 'in a scrutinizing manner'.
You could say of your research, "In the course of my analysis, I had to carefully scrutinize the TV series and transcribe the dialogue".
Best Answer
The word "minutely" does exist, but in the meaning "every minute" it's archaic (see e.g. Merriam-Webster or The American Heritage Dictionary of the English Language). The modern meaning is twofold: "in great detail, in a minute manner, with attention to detail" or "into tiny pieces".
What you are looking for is "minute by minute", "once a minute",
"on a minute-by-minute basis"(thanks to Shinto Sherlock for setting the record straight on this one), or simply "every minute".