Dell Latitude E-series laptops work with out a battery. So even if the battery were dead, the laptop should still turn on if the power adapter and motherboard were in working order.
If you are able to turn on the laptop, press "F2" on the Dell Logo to enter BIOS setup. Sometimes it will tell you what is going on. If the battery is failing it will let you know. Some problems with the AC adapter will also send you a notification in the BIOS.
For the E-Series, if the AC adapter is failing or it is less than 100W or 65W (Depending on the model), it will refuse to charge or charge extremely slow. I see you have a 12" so a 65W should be sufficient to charge. I would say try another adapter first. The next step would be to try another battery.