What details in the story suggest that Miss Brill might be having some health problems and might even be reaching the end of her life?

Our first clue that Miss Brill might be having some health problems is the fact that, as she walked her to park, "She felt a tingling in her hands and arms [...]."  This isn't a particularly normal feeling to have just from walking, as she supposed it to be.  Such a tingling could be a sign of neuropathy (a symptom typically associated with diabetes), and is often considered to be a condition of the elderly.  

Further, as Miss Brill sits on her usual bench at the park, she looks around and notices that everyone around her had "something funny" about them: "The were odd, silent, nearly all old, and from the way they stared they looked as though they'd just come from dark little rooms or even -- even cupboards!"  She notices that everyone around her is old, but it doesn't appear to make her question whether she is old herself.  Moreover, a reader might imagine that going to the park on a Sunday afternoon to listen to a band play is rather an older person's behavior. Her similarity to these "old" people is strengthened by the description of her home in final paragraph.  The narrator tells us that, today, Miss Brill passed by the baker's on her way home, and went straight back to "the dark little room-- her room like a cupboard [...]."  This is precisely the kind of place she believes these "old" people have come from, and we learn, in the end, that she has come from exactly the same kind of place.

Finally, when a young couple sits down next to her, the boy describes Miss Brill as "that stupid old thing at the end there" and suggests that everyone would wish her to "keep her silly old mug at home."  While these statements are unkind, they do show us the way Miss Brill is viewed by someone more youthful.  

This seems to be the moment when Miss Brill understands that she, too, is one of the old people as it is what causes her to break with her typical routine at the baker's.  When she gets home, she takes off the fur fox she'd worn to the park and imagines that she hears it crying when she puts it back into its box.  However, we know that it cannot cry and so the crying must be coming from Miss Brill, and this connects her to the fox which the narrator described earlier in the story.  Miss Brill had

taken it out of its box that afternoon, shaken out the moth-powder, given it a good brush, and rubbed the life back into the dim little eyes.  "What has been happening to me?" said the sad little eyes [....]  But the nose, which was of some black composition, wasn't at all firm.  It must have had a knock somehow.

We might now come to understand that the fox is actually symbolic of Miss Brill.  She, too, is rather dated and musty, having been kept in a little box of her own (her cupboard-like room), with dim eyes that likely signal her age.  While her nose is still attached, it could look quite different in her old age from how it once did; her skin is likely wrinkled and a bit sagged, changing her features from what they were when she was young.

All of these details point to the fact that Miss Brill is likely well into her old age, without even realizing it herself until the very end of the story.  And it seems to be this realization that, in the end, makes her cry.