Three nice set phrases stack here. 居眠り ('dozing off') is specifically nodding off when you should be paying attention — in class, in a meeting. ~分 as a noun means 'the amount/portion of X' — 'the part corresponding to the dozing.' The idiom 一筋縄ではいかない ('can't be handled with a single rope') means 'can't be dealt with by ordinary means, is no easy matter' — a classic fixed expression. The whole 感じが漂う ('a feeling drifts / hangs in the air') turns an abstract intuition into a spatial metaphor. 合わせて 'combining, including' pulls the dozing time into the tally.