| Based on your circumstances, you would simply file your tax return as a Single Filer. The IRS tax code would not permit you to claim the children as dependents as you do not meet the requirements for claiming the children as a dependent.
Your Girlfriend on the other hand would be filing as a Head of Household filing status and would be able to claim the children as her dependents. She also be entitled to claim the child tax credits for both the kids!
Since you cannot claim the child care tax credit, the only option is for your girlfriend to claim the children as her dependents and therefore claim the child tax credit on personal tax return.
If you had qualified to claim the children as your dependents clearly, then the person making the most income would generally benefit by claiming the children as as dependents. But, at times this may not be true as at certain levels of income the taxpayer with the lowest income may qualify for the earned income tax credit.
I suspect that your girlfriend may qualify for the dependency allowance deduction, child care tax credit as well as qualify for the earned income tax credit. | 
