So I did a bit of research to see if I could figure out the reason for the "distance" between these two.

Seems like historically, Shiranui was sunk, with no survivors, trying to save Hayashimo's crew after the latter had been grounded. I guess that's the reason for their relationship.

Fujinami (not yet in the game) also sunk with no survivors trying to save Hayashimo. Choukai's survivors were aboard Fujinami at the time, and also all died. So I guess our little stalker is carrying quite a burden, emotionally.

Or, actually, maybe she's not a stalker after all. Knowing her history, there's another possible interpretation of her idle line about "just watching... forever": She's watching Shiranui and Fujinami sink, unable to do anything about it. Perhaps she can't get the image out of her head, and that's what the line refers to.

Edit: Damn, a bunch of her lines feel different knowing her history. Somebody on the kancolle wikia has a good description:

Hayashimo seems to be more shellshocked than anything, especially if you consider Leyte Gulf. Owing to that she might have a few loose screws rattling around, but that doesn't mean it's anything like Tatsuta or Ooi's flavors of crazy. Like, instead of hearing her idle line delivered by someone holding a knife, picture someone who has stood in the same spot staring out the window all day, unresponsive to offers of going out to lunch or games of Battleship, like her soul has been sucked out through her nose. More accurately, she's probably having flashbacks of being stranded, unable to do anything but watch.

Now I feel terrible for thinking she was a stalker. :(