put a panty hose loosely over the end of the vacuum hose and drag it over the carpet in the area. it will suck up whatever is there, without it ending up in the bag.

the chain more than likely broke. it is the habit of a theif to make it a quick snatch. it would raise questions to see a pendant and not the chain, so why not just sink both in your pocket? even if you had no intentions of hocking the dog pendant because its so specific you could toss it, and hock the chain. its much more likely the chain broke.

When the cleaning lady shows up tell her this is missing. Ask if she has seen it, and if she will help you look for it. it may be under the dresser, or edge of the couch, or behind a cushion. Since you didnt see her take it, you cannot accuse of of the crime. You can suspect all you like, and be prepared in case you think her a thief, such as locking away your valueables, or putting in a nanny cam, or sitting in each room with her while she works, but the best you can hope for is that IF she took it, she will have an attack of conscious and drop it on the floor and say HERE IT IS! or ...I broke it and took it to have it fixed....or I sucked it into the vacuum bag and here it is! covered in dirt.

I hope you find your treasure.