Versions of Windows removed the "Stereo Mix" or "What you hear" options for a number of reasons. The important part is that they're not there any more.
My solution to this problem uses Virtual Audio Cable - sadly not free, but a very useful "device" driver which presents your Windows install with a few new sound devices. A typical setup for a broadcast including teamspeak runs like this (You'll need to configure VAC for a few additional cables):
Teamspeak outputs on Line 1 (VAC)
Line 1 outputs to an Audio Repeater (application included with the VAC pack) and your headset should connect to one output of this audio repeater so you can hear them
Line 1 outputs to another Audio Repeater which inputs to the same repeater your microphone does (We're mixing the audio together because TS doesn't echo your voice)
The above audio repeater connects to your broadcast software so the outside world can hear you all
Your microphone inputs to another audio repeater which inputs to teamspeak (so your co-broadcasters can hear you)
This sounds overly complicated and perhaps can be explained better, but the crux of it is that you must run "Virtual Cables" between the people in teamspeak, yourself, and the broadcast. And because Teamspeak (rightly) does not echo your voice back to you, you'll need to run a cable directly from your microphone to your broadcast software.
You can do it with two cables, if I remember rightly, and about five audio repeaters, which might fall under the scope of the free version. I recommend you fiddle with it until it works, honestly.