I'd think that because she offered the information about going for chemo, she's looking for a bit of support and doesn't mind you knowing she's poorly. Could you pop round with a flask of coffee and cake? She won't have to worry about making you a cuppa, or telling you where everything is which can be tiring. You sound like a lovely neighbour for an elderly person to have.
If she's alone and having regular chemo she may not be up to popping round to visit you, but may very well be delighted at any offers of help with anything in the house, shopping etc. Chemo really knocks you sideways, even if you aren't elderly, so she won't be up to doing much herself.
No offence taken by me - I think I would bite the bullet, knock on the door and thank her personally for the lovely card. She said herself that she's not seen you to talk to, so in my view putting a card through the door wouldn't address that and puts the ball back in her court to make contact which she may not feel strong enough to do. When you speak to her - you'll probably get a good idea whether she'd like a longer chat - you could ask whether she like to walk round to yours for a cup of tea or you could offer to make one for her in her own home if she doesn't mind you coming in. It's really miserable being ill on your own with no one to even make you a cup of tea.
background; we'v been here nearly 3yrs, say 'hello' to this neighbour an older lady,she sent us a congratulations card when dc3 born recently, I thanked her and we spoke a little, exchanged Christmas cards, I happen to know she's had a life-threatening illness as she's a patient at my work and my colleague had her notes open on screen I did NOT open them myself she gets collected by her friend or taxi every day to go somewhere she has a walking frame
SO; the other day she put a card through the door saying sorry shes not seen us to talk to for some time and how lovely we all are esp the dc's and explaining how ill she is and all she does is goes to chemo daily and that its rare to see such a lovely family just the tone of it struck me that maybe she's saying she won't see them grow up sorry if iv offended anyone by putting this in the wrong place id just like some perspective please, what do I do or say now? X