My apologies to Elwyn, the group and PRONI as it has taken me longer to post this than it took for them to respond.
As previously mentioned I used the e-catalogue references kindly provided by Elwyn to find the resources which contained personal records for patients.
I then used the online query form at
https://www.nidirect.gov.uk/articles/proni-enquiry-service and submitted a request for records relating to my gg grandmother. In this I included the links to the civil marriage and death records available at
https://www.irishgenealogy.ie/en/ and the relevant reference numbers and received an initial response a week after submitting.
Twenty days later I received the information that they had managed to find relating to my query.
Sadly in my case they were only able to find the entry relating to her death, but this did contain the date that she was admitted to the Asylum indicating that she had been released from Peebles Asylum on the 3rd June and admitted to the Londonderry Asylum the next day.
Definitely worthwhile trying due to the very prompt response from the Team at PRONI.