How do I tell an ex I can’t speak to her anymore, after she asked to stay in touch after breaking up with me?

Tell her staying in touch is not working for you and you need to go "no contact" from now on. Setting a boundary with someone may be uncomfortable but it's you taking care of yourself. You are not obligated to talk to an ex even if you initially agreed to after a break-up.

I think it's best to be as direct as possible. However if you think your words my hurt her, then explain to her that you both need time to fully move on and that it's best that you don't talk with each other to do that. The pain of breaking up would be still fresh and talking to each other would just never let yourself recover. So explain that perhaps?

Hmmm, well, tell her that you need space and that it might be a while before you speak to her again.

Your absolute best call is to be completely honest and straightforward about it, as beating around the bush won't do much except make things harder later on. Chances are, if you don't want to stay in touch now, you won't want to later on either, so it's better to do it sooner. Consider why you would prefer not talking to her, then tell her exactly how you feel, why you feel this way, and why you think it would be better not to stay in touch. Taking care of yourself and your personal comfort comes first, regardless if it might hurt her.

You need to be as direct as possible, if you are even remotely convincing she will continue to contact you, even if you block her she will find a new way to talk to you, Having a direct conversation and clearly stating what you want. For example "I know you want to stay in touch, but I can not speak to you anymore, it does neither one of us any good speaking to each other, so we both need to move on with our lives and if we cross paths again one day cool, but right here right now no more speaking".

I would just be upfront with your ex. Explain that although you may have nothing against her as a person, you are emotionally unable to "stay in touch" with her during this time. I'm sure she will understand how difficult it can be to heal or move on when you're still in contact with the person who broke up with you. You could suggest that maybe one day you two could keep in contact and have a friendship, but for right now, you're unable to play that role in her life. Be clear about your boundaries and hopefully she understands.