She pronounced it as せんたっき, not せんたき. According to NHK, this is an acceptable pronunciation of the word. The く was not removed, but replaced with the small tsu!
This is a phenomenon called 促音化 (gemination). You probably already know that 学校 is formally spelled and pronounced as がっこう instead of がくこう.
せんたっき is in the same vein, but it's still in the middle of the full gemination process. That is, all dictionaries and most people say the correct spelling is せんたくき, but more than half of the people actually pronounce it as せんたっき almost unconsciously. (But せんたくき is still correct, of course.)
You can read about gemination in the following articles:
This is a similar phenomenon.
(By the way, if you missed the small tsu that was there, that means you need to train your ears! せんたき and せんたっき are as different as night and day to Japanese ears, and she pronounced it really clearly in the video.)