First, Miss Brill doesn't know she is a lonely old woman until the Romeo and Juliet couple in the park tell her she is. Her life has been rather pleasant up until this encounter with Romeo and Juliet; it has been rather like her Sunday honeycake that generates great anticipation and "a dashing way" to her life:
Sometimes there was an almond in her slice, sometimes not. It made a great difference. If there was an almond it was like carrying home a tiny present ... She hurried on the almond Sundays and struck the match for the kettle in quite a dashing way.
Second, we are told very little about any of her circumstances but we do know:
- she feels pleasantly happy at all times, which is confirmed by her dedicated attention to her beloved fox.
- she has two costly and elegant possessions that we know about: the fox fur necklet and the Eider duck down eiderdown comforter, coveted by luxury lovers.
- her room is a small one as the red eiderdown is before her as soon as she walks in:
she ... climbed the stairs, went into the little dark room–her room like a cupboard–and sat down on the red eiderdown.
We can make inferences from these random bits of information. We can infer that she once had a more elegant life and that she had some level of social prestige. We can infer this because of her fox and red eiderdown, which are each costly and a suitable part of a less constrain lifestyle.
These puzzle pieces don't provide much of a picture from which to answer your question but the cumulated inference is that once, she lived differently until some unknown circumstance occured (perhaps the death of her parents and loss of wealth or the death or loss of her fiance ...) that reduced her situation to near poverty, though she took it bravely and kept up her spirit and good cheer.
She may have cherished a secret hope that at sometime, things might change and she might regain her former position. With such a secretly cherished hope, time passes without being noticed and the remarks of Romeo and Juliet might have shattered the mental images of her life that she had frozen in time. So she may have become (not been) a lonely old woman that day because she recognized the reality of her life and future or because her secret hope was shattered.