I doubt that you are still looking for the answer, but if you are, here's what I found:
If there had been any doubt about French's acting ability, this had been dispelled the previous year, 1993, in the BBC Screen One drama Tender Loving Care. In this work, French played a night nurse in the geriatric ward of a hospital. There she helps many of her charges "on their way" with her own brand of tender loving care, believing that by killing them she is doing them a service. It was a beautifully understated and restrained performance.
http://www.museum.tv/eotv/frenchdawn.htm
Screen One: Season 5, Episode 5
Tender Loving Care (6 Oct. 1998)
TV Episode | 75 min | Comedy, Crime, Drama
Dawn French plays a nurse who murders patients on her hospital ward.
http://www.imdb.com/title/tt0143927/
Here's are links to websites that list Murder Most Horrid episodes:
http://epguides.com/murdermosthorrid/guide.shtml
https://en.m.wikipedia.org/wiki/Murder_Most_Horrid
I looked quickly but I didn't find a similar synopsis to what you describe.
"It's a real burden being right so often." Captain Malcolm Reynolds, Firefly
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