You could likely just solder a wall wart with compatible voltage  to the battery connections on the case of the computer.

Depending on what you have for warts laying around though. 

What model lenovo is this?

Ideapad S9. I wanna just start slinging voltage at the battery pack, but I also don't want it to blow up in my face

Type: 6 cell lithium-ion

Output voltage: 10.8 V dc (nominal)
Capacity: 62 watt hours
Charge time (power off): 4 hours
Color: Black

That came off the datasheet. Make sure you don't leave that unattended, If you want to do it as you say I would check it every few minutes. Overcharging a lithium is a fun event.

In most cases the powersupply socket has been yanked loose on the motherboard. It is a simple soldering fix ... once you get to the spot.

Find and download the excellent Hardware Maintenance Manual for your model (Oh, I guess that might be just for Thinkpads)

Or find a teardown video for your model on Youtube.

That would restore the laptop to full functionality.

I have such an adapter, possibly someone near you might as well. You'd also need another hard drive to put the data onto. My point is that there is another way to get the data, and perhaps posting something about laptop hard drive data recovery, and your general location (details when needed), might get you a solution?

You are correct, I could easily recover the data another way. I have the skills necessary as well as the spare computers. Really I wanted to go this route because I was hoping to reuse the laptop as well.

Resolder the friggin' power-connector to the mobo, dag-nabbit! Isn't that what you said was wrong?

Yeah, although I'm not sure if that's the problem. The voltage isn't getting to the battery for some reason, I hope it is just the jack, in which case soldering straight to the board will have me with a working laptop again. Although it will have a permanently attached cord. But that's the way the news goes

I assume the files are on a 2.5" IDE HDD so what about http://www.amazon.com/gp/product/B004ID6J0K?keywords=2.5" ide adapter&qid=1445615397&ref_=sr_1_2&sr=8-2 or http://www.amazon.com/gp/product/B000J01I1G?keywords=2.5" ide adapter&qid=1445615397&ref_=sr_1_1&sr=8-1? The second one is about as much as a dock but I find it quite useful from time to time.

If you still don't want to spend any money and want to build an adapter yourself I could send you some images of the internal connections of that 2.5" to 3.5" IDE adapter (which are almost 1-to-1).

I don't get why the laptop wouldn't work without the battery and only with the psu connected. What happens when you try that? 

It won't work that way. I'm thinking the connector on the board is fried or something similar. Although it's putting out voltage, it's not going anywhere

Ahh, that makes sense!

I think if the data is not important, do not have to be restored

Can you extract the hard drive? It would a LOT easier to recover data that way.

It's an older laptop, but I don't wanna spend any money to recover the data. I think I would have to get an adapter or something to use the drive

Clill it for some hours and try the battey again

Or:

May give a short boost, save files as soon as possible

Great idea and one lots of people could use, but yea, if your pressed for time or simply anxious buy/order a 2.5" HD dock, some of the better ones (mines was on sale for under 20 bux) accept both the new sata as well as older ide HD's, anyhow, good luck.

You could just pull the drive, you know.

Well, #Smart Battery, Smarter Power Pack might just be the project you're looking for ;) ! Let me know if it can work out.