In the video "Eating disorders: Why is it so Hard to Treat Them? | Ilona Kajokiene | TEDxVilnius"(https://m.youtube.com/watch?v=vGFvn37tRbM), what are the main points of connection with the Blythe Baird poem "When a Fat Girl Gets Skinny," and what are the points that offer new information? What quote of the speech do you think is key?

That's a solid video, and it definitely relates to Baird's unflinchingly honest poem. It's why I use both of them in my high school health class. The TED video is a great clinical look at eating disorders, and the Baird poem nicely ties in from a much more emotional viewpoint. Despite the different angles of approach to the dangers and consequences of eating disorders, the TED Talk and the poem have a solid amount of overlap between them. That's what makes them so valuable as a pair. Each highlights the other.
As previously stated, both sources specifically discuss anorexia. Both sources also talk about the disorder as a type of mental illness:

How could I not fall in love with my illness?With becoming the kind of silhouette people are supposed to fall in love with?Why would I ever want to stop being hungry, when anorexia was the most interesting thing about me?

Baird's poem doesn't explicitly state "mental illness," but she comments often about her eating habits making her happy or guilty feeling:

Watching Americas Next Topmodel like the gospel,hunching naked over a bathroom scale shrine,crying into an empty bowl of cocoa puffsbecause I only feel pretty when I'm hungry.

The TED video and the poem both discuss eating disorders as a result of feeling body shame from one source or another. It could be media influence, family influence, or peer group influence. Finally, both are centered on females. The TED Talk points out that most eating disorder patients are female. Baird happens to be a female talking about the issue, but I do want to point out that the peers who congratulate her are female and want to know what she is doing to find weight loss success​:

So when I evaporated, of course everyone congratulated me on getting healthy.Girls at school who never spoke to me before, stopped me in the hallway to ask how I did it.I say "I am sick". They say "No, you're an inspiration!"

Once the TED Talk video reaches about the halfway mark, viewers start getting information that the poem doesn't go into. The video begins to provide ideas about eating disorder causes that are backed up and supported by medical research and years of psychological work. One key point that the presentation makes is that there is a link between the biological and psychological aspects of eating disorders as it specifically relates to the causes of the disorders. It's not one or the other. They are intricately linked and woven together. This is an interesting connection because it means that some people are biologically​ predisposed to ending up with an eating disorder. Kajokiene then spends just about the final third of the video breaking down three specific "factors" that contribute to eating disorders.
The first factor Kajokiene names is a "predisposing factor." This factor is completely a biological factor. It is genetically determined. Our genes determine a great deal about how we look. Things such as body type, height, build, and the like are genetically controlled factors. Interestingly, Kajokiene points out that certain genetic traits give people the ability to ignore for longer periods of time the hunger drive. People with anorexia can quite literally starve themselves longer because they are genetically predisposed to be able to do it.
The second factor is called the "triggering factor." These are environmental factors. It could be something as simple as somebody making a comment about your weight. It could also be stress related. The point is that there is some environmental trigger that releases the hidden biological predisposing factor.
The third factor is the "maintaining factor," and this is the factor that makes it extremely difficult to stop a person's eating disorder. The maintaining factor occurs when somebody makes a comment about how good you look because of how thin that you are. Your thinness is being affirmed, so you feel good about what you have done to get that way. It's a positive feedback loop, and people don't want to stop their eating disorder because the positive compliments will perhaps stop. The Baird poem specifically mentions this factor. The poem doesn't name it, but it does describe it quite nicely:

So when I evaporated, of course everyone congratulated me on getting healthy.Girls at school who never spoke to me before, stopped me in the hallway to ask how I did it.I say "I am sick". They say "No, you're an inspiration!"

As for a "key quote" from the TED Talk, that is left up to the individual viewer; however, I do think that an extremely important piece of the presentation is that a person's eating disorder "slowly becomes an identity . . . and getting treatment means losing that identity." This means that when a patient gets treated for an eating disorder, that patient is literally asking "If I will be treated, who will I be then?" For me, this is a key quote that hauntingly refers back to Baird's poem and the part that Baird mentions falling in love with the disease because it is the most interesting thing about her:

How could I not fall in love with my illness?With becoming the kind of silhouette people are supposed to fall in love with?Why would I ever want to stop being hungry, when anorexia was the most interesting thing about me?
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3010958/