We just returned yesterday from our second trip to Mexico (not Mexico City, but for the most part, it's the same around the country when it comes to water). On this second trip, we did get sick. We followed the same precautions as last time: We only ate at well-reviewed places, only drank filtered water, and brushed teeth with tap water.
We didn't get sick until the end of our 4 night trip. You should be fine ordering ice -- from what I've read, Mexicans all buy bagged ice from the factory, and even Mexicans drink filtered water, so they would not serve you tap water (or ice made from the tap). I would honestly still just avoid ordering from someone who looks like they may have sanitary practices that my stomach may not handle (e.g. the vendor who leaves the tub of mayonnaise out in the sun all day).
If you want to play it safe, follow the other tips at avoiding street vendors, and brush teeth with filtered water. I've read 50/50 mixed reviews about whether it's okay to brush with tap water, so we have chanced it, but in the future we will use filtered for teeth brushing. We didn't get "very sick", only mildly so, so it's possible this was our mistake.