Re-seat the batteries. The blinking red light means it is not charging. The blinking light means that an incorrect amperage/current is present. I have fixed a number of these with that symptom by removing the battery and putting it back in. If this doesn't work try the following on the XL.
On the DSi XL since you have 2 of them. Exchange the batteries between the 2. On the system currently working try charging the battery from the other system. That will tell you if the battery is any good. On the system not currently working try turning it on with the known good battery in it. Also try charging the known good battery in it. That will tell you if there is more problems than just the battery with it. If this doesn't solve your problem try oldturkey's advice.