Thanks for all the great contributions, guys! This thread is really coming along nicely!
I've been listening to a lot of Fujitani Miki lately.
That was a cover of a classic 70s song - Ohta Hiromi's Momen no Handkerchief. Here's the original:
Ohta Hiromi is a legend, so I highly recommend you listen to her albums as well if you haven't already!