The death cert may or may not help; it will give the place of death, but the informant may be someone from the hospital rather than a relative.
If the 1911 entry is the correct one, which I'm afraid looks very likely, this is probably where she died:
http://en.wikipedia.org/wiki/High_Royds_Hospital
You can find brief details of the Wakefield and Menston asylums here:
http://studymore.org.uk/4_13_ta.htm
And this database shows what records are available, and where to get them:
http://www.nationalarchives.gov.uk/h...rchresults.asp
Some records have a 100-year closure period, but I think some are 70 years. I had no trouble getting the asylum file for a great-aunt of mine who died in an asylum in 1930.
You can get a transcript of Menston Hospital burials quite cheaply here:
http://www.parishchest.com/menston_h...0-1969__P80138
but I don't know if it'll help much (you'll probably only get the person's name and date of burial, and their abode).
There were many reasons why someone might be in an asylum, but the files I've seen so far showed elderly people with symptoms that sounded like dementia, or younger people suffering from something like schizophrenia.