Very sad Vietnamese coriander

I have a couple of Vietnamese coriander plants from last year that have been inside all Winter on a window sill. About a month ago I noticed some sticky deposit on some of the leaves. Over the last 10 days the leaves have suddenly started to turn red and then brown. I don't know if this is a nutrition problem,.or a parasite problem, or both! I don't see any whitefly or greenfly but I'm definitely far from expert here.

Can anyone advise what to try?