What you both are running into is a form of content protection coming from DirecTV in the form of SOC (Selectable Output Control). What this means is the DirecTV box detects an HDMI connection, so it shuts off the video signal for all the other outputs (such as the 10-pin cable you're using for the Slingbox). This article mentions the issue (the section called "What is Selectable Output Control (SOC)?"):
There are a couple workarounds for this issue:
1) Disconnecting the HDMI cable altogether should bring you your video signal back (reboot the cable box if it doesn't). Then, in order to bring the signal to your TV, you simply connect the component cables that came with the Slingbox (or a spare set) from the Slingbox OUT to the TV.
2) If you wish/need to keep using your TV with HDMI cables, we recommend the use of an HDMI to component converter. Here is some information on the one we recommend:
Hope this helps!