Sorry to hear that cappucino.
My mum is still very much alive but we didn't really have a mother-daughter relationship either and I was very much like the film Carrie when I started mine.
My advice is to always be open and honest. My dd knows all about my periods. She has seen me change my pads from being very young. They learn shame, they are not born with it and so I've never taught them to feel ashamed or embarrassed of their own bodies. That can be hard when you have learnt shame yourself, but when they fling open the bathroom door as I'm changing my pad and then carry on as if nothing unusual is happening, you soon get used to it!
Because dd has always known about my periods, she's asked lots of questions about whether or not it hurts, how long it lasts, why it happens, etc. However you can get a book about all of this and that might be a good idea if you feel uncomfortable talking to her about it.
I would start off by showing her what you use and telling her a bit about your own periods. You can share your teenage experiences with her - have a laugh about it to break the ice. I'm sure you must have a funny story to tell her. Mine was about a girl at school who put her pad sticky side up as she thought it had to stick to your vagina rather than your knickers!
Show her the lillets website and ask if she'd like to try them for when she is swimming. Tell her how you felt about them when you were a teen. You can also show her the mooncup - her first reaction will probably be one of horror and she'll be grossed out by it, but then again you can have a laugh with her about it and joke about how on earth women manage to get it up there. She might dismiss the idea straight away but you will have piqued her curiosity and hopefully by having a laugh with her and a friendly chat, she'll be more relaxed about opening up to you in the future.
HTH 