As for why Congress has always been dragging feet in re-imbursing the border states for the cost that they incur, this mainly boils down to congressmen from other states not wanting to spend the money. So as the illegals show up around the country, they will want their share also. If Texas gets re-imbursed, GA and NJ will want to be re-imbursed also.
Do would these funds get disbursed directly from the treasury to the states, or would they go through DHS? Is it possible that DeLay was trying to find out what the "delay" was in getting these funds to their destination?