The problem I have is that living and working here, I think it's a basket case economically and really struggle to bring myself to invest here.

It is a common concern, but there can still be very good firms in a struggling economy. Very good firms outlast the struggles, one hopes.

It does seem tricky, but I tried the FT.com global equity screener.
A list of industries which excludes the usual duds and high-risk things
Five year average ROAA > 15%
Five year average ROE > 20%
P/E < 22
Debt to total capital < 40%

That left only 5 US listed firms, alas.
4imprint Group PLCFOUR:LSE (media)
Bioventix PLCBVXP:LSE (pharma)
Gaming Realms PLCGMR:LSE (leisure)
Kainos Group PLCKNOS:LSE (tech)
MONY Group PLCMONY:LSE (tech)

I don't know anything about any of them, other than what I fed into the screener. But their site has an info page for each, as a start.
e.g., Financial Times: Kainos Group PLC, KNOS:LSE summary
That one looks pretty good at first glance.

Jim