Is it rude to say “my memory is not even good,haha.” as a reply to an older person?

I don’t find it rude. In fact, it seems pretty tactful and funny. She probably just forgot again.

Nothing about it was inappropriate but the phrasing was a bit strange. I would say “my memory also fails me sometimes” or something like that. However I would personally just ignore her saying this or just laugh as your response obviously references her older age which is a no-go usually. Nothing you have done is crazy rude, but a slightly politer response could have been possible. Don’t beat yourself up about it; you live and you learn.

In written communication, especially among co-workers who don't know each other very well outside of professional interactions, there's always the chance that someone might read an offensive tone or inflection into what you say. Of course, it can happen in verbal/in-person communication too, but you have more tools at your disposal (*actual* tone, body language, laughing, etc.) in those cases to make your intentions more clear. Especially in professional communications like this, it can be a good idea to be very careful with what you type. That having been said, outside of any context other than what you've shared, I don't see anything wrong with what you typed. You didn't say or even imply anything about her. You said your memory isn't good, which I think I'd read as you essentially saying "You don't need to apologize for a memory lapse, as my memory's bad too. So I understand." It sounds to me like you were trying to relate to her and joke around to express a bit of kinship, not saying anything offensive. But, going back to my first paragraph, I don't know anything about this woman or what other communications you've had with one another. Is it possible she took it wrong? I can't say it's not possible. I can say, though, that there's probably no reason for you to interpret her lack of response as her being offended. She may have gotten busy, or she could have considered the communication complete. Or, as we've established, it's possible she forgot. Again.

I don't find it rude either, hardly any time has passed, and she is probably just busy. Try not to let your concern about other people’s feelings let you imagine that they are upset with you :)

Oh no! I definitely feel that it was not the best to say that. Thank you everyone for sharing your thoughts and making me learn!!

In British English, say: “no problem, I’d forget my head if it wasn’t screwed on”. Or “no problem, I’ve got a memory like a sieve.” In British English, you can show your sympathy for someone by saying that you are the same, in a self-deprecating way. (= saying bad things about yourself.). But, I’m not sure if this would work in American English, Australian English etc. what would Trump say? ‘We shouldn’t have employed mentally defective people.” ?

I don't think your response is the reason she isn't responding. That said, there are probably better ways you could have phrased it. But if she took that the wrong way it's more on her than it's on you.

I think she probably forgot to respond to your email lmao. Also, your email is absolutely not rude, but it's not phrased in the most correct way. The first sentence is great (although I would put an "it's" in front of it as a matter of personal taste more than anything else) but I would phrase the second sentence as "I don't have a good memory either, haha" or "I don't even have a good memory, haha"

If it took her a few weeks to read and reply to your first email, she probably hasn’t even seen your second one yet. If it makes you feel better I chuckled at your reply. I would not have been offended at all.

A lot of native English speakers are going to realize English is not your first language and will be able to understand your meaning. I work in tech and come across this type of thing regularly. She's not mad, but she probably does have an issue responding to emails in an expedient manner.

Your grammar is wrong. Correct constructions world be: “My memory is not good either”. “Mine is also not good” Or any of the options others have given you. But I can’t imagine that’s the reason you’ve not heard back.

I think that's universally OK or even nice to say across cultures. We don't speak English and we use same tactics to comfort someone from regret, too.