Dells commonly have this problem. It's the power connector a lot of the time, but you should make every effort to isolate the power brick as the cause before scrapping it out as they are also a commonly defective product over time.
If the laptop actually has a broken power connector you can sometimes buy a new connector online for like $5 and replace it yourself. It's kind of a chore since you have to dismantle the laptop and carefully unsolder the old connector and solder a new one in it's place.
Sometimes the Power Connector is separate from the motherboard and connects to it via a 3-wire header. Those are the best because the soldering is eliminated. I'm not sure I've ever seen that on a Dell.
The other route, as has been mentioned, is to find a docking station if your laptop can use one, and if you can find one cheap enough.
But yes, your question about using a portable enclosure to access the drive is correct. That would work great to recover your files and then you can use the drive as portable media, or as a backup destination, or both.
If you provide the model number I can search for instructions on repairing it. A quick lookup for the part will tell if it's the easy to fix pigtail connector, or the much more difficult soldering style. It will also tell if your model will accept a docking station.