Help with sick Kentia Palm

[LukeFl]

I got my Kentia Palm (as is typical, multiple plants in one pot) in September.  It's mostly been okay, but the last few weeks have seen some problems, It seems at least one of the specimens is very sick, sadly. It's gone brown and dry in parts.

I've been watering every week, in almost every case the top inch was dry before I watered. I've sometimes been sip watering - just giving a glug at a time - which I now know to be incorrect. I fertilised once or twice with organic sea shell fertiliser plus a bit of used coffee grounds.

The plant's fronds had black spots on the underside even when healthy. I did look them up when I first got it and something I read made me think they were okay, but it seems like they might be signs of a fungus or even a parasite. They don't seem like insects though but just tiny black grains, like coffee. The thing is, they seem to concentrate on the underside of the sickest areas, which makes me concerned they're involved in the plant's ill health somehow.

Any help much appreciated!