Questions, questions....
I'm only going to discuss Federal tax and credits, because, although DC may have it's own income tax and employment tax rules and credits, I have no idea what they may be.
If she lived in DC less than 39 weeks, the moving expenses would not be deductible by her (him?) if
she paid it, so it's includable (on her income tax returns) when
you pay it. (The moving expenses
back are not deductible or excludible in any case)She is required to report the moving expenses (as are you on the W-2, if otherwise required to file a W-2), so you can clearly include that as eligible expenses. You can also include as eligible expenses, meals and a portion of utilities, according to Pub 503.
In California, if you would otherwise be required to file State employment tax returns, you would include meals and 2/3 of the Fair Market Value rental of her room, in addition to wages and the moving expenses, as being subject to employment taxes.