I recently resolved a client's Office 2013 installation problem and wanted to offer an observation that may help you with troubleshooting similar issues.
The client ordered three Dell Optiplex system units with the Windows 7 downgrade option installed. They set up a trial account for Office 365 and installed on the first PC without incident. The second PC went through the installation process but would not start any of the Office programs.
The decision was made to forgo the Office 365 and the client purchased licenses for Home and Business Office 2013. The third computer had this installed without incident. The second PC again balked. The Microsoft troubleshooting path of using the Fixit
to uninstall Office and then repeating the install offered no joy. A full manual uninstall was performed. No joy. The "Something went wrong. Sorry, we ran into a problem. Try going online for additional help."
message continued to appear shortly after initiating the install, and once the install declared itself complete, none of the programs would open, and the dreaded instruction "We couldn't start your program. Please try starting it again. if it wont start, try repairing Office from "program and features" in the control panel."
As a matter of expediency, Office 2013 Standard from a volume license was installed without activating so the user could get to work. Evidently something in the handshaking around the streaming Click2Run process was failing.
When an opportunity presented itself while the user was away for a few days, I did a complete manual uninstall, attempted the Home & Business install again, it failed, and I contacted Microsoft.
Level 1 tech support repeated all the steps described above with similar no-joy results, and kicked the problem to Level 2. The Level 2 tech attempted to repair the OS remotely, but this led to an bricked PC -- the keyboard and mouse would not respond once the license key screen appeared.
I went on-site and created a recovery USB stick from computer 3 (the DVDs from Dell were for W 8.1, not W7), and used that to attempt a recovery using the advanced tools (essentially the Microsoft recovery menu). This didn't change the situation at all so I had the Dell Quick Restore run (what the hell, it was bricked at this point). That worked. The system came back up, I was able to install Home & Business Office 2013 (the suite version, not Office 365) without issues.
Long story short: If the full manual uninstall doesn't work, reinstall the operating system. It's not worth fighting the problem past that step.