There used to be a recording application that you could get. Sadly, I don't think it works anymore. It was called Skype free recorder or something like that. You could always try something like VBCable. https://download.vb-audio.com/Download_ … Pack43.zip
Before doing anything, I recommend setting your screen reader's output device to your default sound card instead of Microsoft Sound Mapper. This will ensure that you do not lose speech if things go wrong. Set Skype to output through the VBCable output device and set your microphone to go through the VBCable input device using the listen feature of Windows sound settings. Then just set the input portion of VB Cable to output to your default sound card using the listen feature so you can hear Skype. Now open your recording application of choice and select the VB Cable device as your recording source. It should record Skype and your microphone. You'll have to listen to yourself, but that's a small price to pay.
If you have Windows 10, you can use the cool new sound mixer to give you even more control. You can now specify an input and output device for each application individually. Windows 10 keeps getting better and better!
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You may access my NVDA Remote, Three-D Velocity, Sound RTS, and Road to Rage servers by using the address christopherw.me. Road to Rage uses the default 6789 port.