Hello Deb, and welcome to RootsChat
In 1994 some of the Littlemore Hospital records were transcribed, and I have an ancient copy of the CD issued by the Oxfordshire Family History Society. Here's all that I found about Rebecca.
22 May 1854 Admitted to Littlemore, patient no. 1002.
(n.b. this might not be her first admission as new numbers were issued on re-admission)6 Jan 1864 Transferred (with many others) to Worcester County Asylum at Powick.
(This was common practice when hospitals were overcrowded)4 Mar 1896 Buried at Littlemore Hospital, age 88, last abode Bicester.
Sadly that's all I could piece together from the CD. Details of patient deaths were transcribed up to 1893 only, presumably because of the 100-year rule, although all of the burials were included. Also none of the admission registers were transcribed at all, so we don't know when she returned from Powick.
Do you have her death certificate which would show her cause of death? Did you know that her hospital records should be available from the Oxfordshire Health Archives? I obtained a copy of my g-grandfather's Littlemore records years ago for a modest copying fee - well worth it.
Must stop now as it's stupidly late (early!) here.
Carol