The battery is not charging from your description.
1. power off, remove battery and put on charger and then try again. battery shorts are common on the D620.
2. try another power adapter as the power adapters have a pretty high failure rate on the D series Latitude laptops (I have over 120 of them at my office, they are not very reliable and have many issues)
3. check and clean battery and power contacts
4. check power connectors to mb and to batt
update if these do not work