My DD has a speech impediment and, although speech therapy is helping, she still has trouble with some words. Even at an early age, if someone would laugh (because it was cute) she would get upset. Your DD is gonna start wondering why Grandma laughs at everything that comes out of her mouth, because little kids can be just as serious as an adult, no matter how cute it seems, and to laugh at them seems to discount their feelings. I would also have an issue with your MIL laughing over house rules, because they aren't funny, they are there for a reason and to me, laughing seems like (not that I think she is) discounting your authority and rules. Maybe you can gently say "Mom, when you laugh at everything she says or does, she doesn't think you are really listening to what she is telling you. Can you please have a regular conversation and listen to if she is being serious or silly?" She is just so probably thrilled that her granddaughter exists that she just has joy over everything, without listening (I have one of those too!) Good Luck
POD to this, especially the bolded. I don't think MIL is necessarily trying to do anything bad, but considering that DD not only gave her a weird look but also walked away from the conversation, the constant over-the-top laughter doesn't seem to be engaging her. Also, I don't think another adult laughing at your house rules is very polite. If this happened frequently it could teach DD not to take them seriously.
Being non-confrontational, I would probably try to get around it without addressing MIL directly. As people suggested earlier, maybe try to guide them through an activity together, like reading the same book or singing a song, where long bursts of laughter would be more obviously disruptive.
Or I might say something like, "Lately we've been working with DD to focus on having an actual conversation of several lines. It's so hard, though, because I have to remind myself not to laugh at the adorable things she says! But I've noticed that just distracts her." And see if MIL gets the hint.
Or, maybe modeling how you'd like MIL to respond--the next time she laughs when DD repeats a house rule, for example, pretend MIL isn't laughing and say to DD (so MIL can hear), "Yes, that's one of our rules, isn't it? That's a very important rule in our house. I'm so glad you remembered it." (Or whatever you would normally say to DD.)
As for repeating the joke over and over to other people... If she's doing this during her conversation time with DD, you could approach it from the "distraction" angle again--DD's attention span is limited, so when MIL makes these asides, DD loses focus and doesn't want to talk to her. If she's trying to get you and your DH to laugh, just stare at her blankly. "Yes, we heard what DD said. I don't think she was trying to be funny, though." Maybe tell FIL that he shouldn't be pressured to laugh if he doesn't feel like it--that you won't be offended, for example. And if you're with MIL at a later time when she retells the "hilarious" story, feel free to just kind of shrug at the other person, so they don't feel pressured to laugh.