You can use the Honorbuddy console to output any information you need.
Running this into console:
PHP:
Logging.Write("-----");
Logging.Write("Area ID: {0}", Me.AreaId);
Logging.Write("Zone ID: {0}", Me.ZoneId);
Logging.Write("Map ID: {0}", Me.MapId);
Logging.Write("Hearth ID: {0}", Me.HearthstoneAreaId);
Logging.Write("-----");
Would output this onto Honorbuddy if you're in Stormwind (and your Hearth is set to the Shrine):
PHP:
-----
Area ID: 301
Zone ID: 1519
Map ID: 0
Hearth ID: 6484
-----
Of course to get the IDs you need, you'd need to be in the location you're wanting the IDs for.
According to the documentation, the 'DungeonID' is the same as
MapId. So, that's what you'll need from the above code.