You say that she was born in January. There are only two Elizabeth Oakley's born in that quarter:-
OAKLEY, ELIZABETH WILLETTS
GRO Reference: 1881 M Quarter in WOLVERHAMPTON Volume 06B Page 621
OAKLEY, ELIZABETH SARAH WHITE
GRO Reference: 1881 M Quarter in BATH Volume 05C Page 614
If she was in the Mendip Asylum that would suggest that she is the one born in Bath. Can you confirm that?
If she is Elizabeth Sarah, then this could be her death:-
Deaths Mar 1959
OAKLEY Elizabeth S 78 Wells 7c 306
Wells is in Somerset.