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Not sure where else to look, sadly, Jackle, if the Archives Department mentioned earlier has no records for her.

You could look for her in Electoral Records ahead of 1939 if you've not already done so, which might pin down where she was living - but that still won't confirm when and why she was in the hospital.

If you google rheumatoid arthritis, the NHS site does give some details about complications which can be life threatening, although I would have thought those might be specifically mentioned on the death certificate, with the arthritis cited as being behind them.

You could also look in local newspapers to see if there is a death notice - I've had a quick look in the British Newspaper Archive but cannot see her.

http://www.workhouses.org.uk/Fylde/

And this, at the foot of the page shows some pictures.  The Institute may have been the infirmary?

What does the death certificate give as cause of death?

"Records can be found at: Archives Department, Postgraduate Medical Library, Victoria Hospital" including "Clinical & Patients 1881-1966".

http://www.nationalarchives.gov.uk/hospitalrecords/details.asp?id=1689&page=69

Is this his wife's burial

Burial: 24 Aug 1897 St Saviour, Bamber Bridge, Lancashire, England
Margaret Beardsworth -
    Age: 40 years
    Abode: Bamber Bridge
    Buried by: George Hilton Curate

the burial records for St Saviour on lan-opc only go up to 1900 but there are quite a few other Beardsworth burials there pre 1900

Is it Beadsworth or Beardsworth? 

Have you found him on the 1911 census to see what his situation is?  Married, occupation etc