Peter,
If installed from a command windows as per the example, the installation log (archydrotoolsInstall.log) would be in the same folder from where the installation package (ArcHydroTools.msi) was run.
However, it looks like something is wrong as your screen shot of your "Toolbox Window" does not show Arc Hydro Tools, yet its .tbx file is visible in your System Toolbox screen shot. I am pretty sure that when correctly registered the Arc Hydro Tools toolbox is treated as a "default" toolbox and should show.
On the other hand, the HEC GeoHMS Tools does not get picked up in the Toolbox Window--so it is not treated as a default toolbox and must be added. Adding it to the Toolbox Window should proceed without issue--as is visible in the attached screen clip where the GeoHMS Toolbox has been added.
So I suspect your installation did not run completely, as the screen capture of your Add Toolbox dialog does not show the GeoHMS Tools as present in the System Toolboxes object.
Have a look in the C:\Program Files (x86)\ArcGIS\Desktop10.0\ArcToolbox\Toolboxes folder for the GeoHMS Tools.tbx.
If not there, something went wrong with your installation and I'd look for problems with your installer privileges and reinstall.
If the toolbox file is present, then I would rename or delete your ArcGIS user profile (the files listed below) to allow default per user settings to be reestablished for your account on next launch and see if working with a default profile clears issue:
[INDENT]C:\Users\<username>\AppData\Roaming\ESRI
C:\Users\<username>\Documents\ArcGIS\Toolbox.tbx
C:\Users\<username>\Documents\ArcGIS\Default.gdb **careful of deleting this one as you may have project data held there
and from a Registry edit session:
rename or delete the HKEY_CURRENT_USER\Software\ESRI[/INDENT]
If the GeoHMS Tools.tbx is not present in the Toolboxes folder, then the installation was flawed. Reviewing the log file may help you to identify the issue. One common problem for example would have been to run the installation as a non-priviledged user. The log should help identify that as well and I'd first check to be sure you've run the installation from a command window "Run as Administrator".
Stuart