togo
You should be able to connect a monitor to the laptop with the broken screen.
If that's not feasible, you can pull the drive and put it in an enclosure to make it an external hard drive or use a docking station like this one.
https://www.amazon.com/Thermaltake-BlacX-N0028USU-Docking-Station/dp/B0012Z3MKW
Don't swap the drives and try to boot another machine with it. It probably won't work, and you could corrupt the drive.
The previous suggestions are good ones!
1. The external monitor trick is an easy solution, should work well, and give you plenty of time to back your Lenovo data up to an external USB drive. Plus, since your system still boots up it's not yet time for more extreme measures. This would also probably be a good time to make a system image with Macrium for safe keeping of all your data.
If booting was an issue, I would proceed with the next step.
2. Use a USB external enclosure or dock for the Lenovo drive.
I like this one, by Inatek. It works with 2.5", 3.5", and 5.25" drives, plus it's SATA and IDE compatible. This is a great tool to have for copying data to another drive or system.
https://www.amazon.com/Inateck-Converter-Adapter-Optical-UA2001/dp/B00MLYV5XY/ref=sr_1_13?s=pc&ie=UTF8&qid=1507218287&sr=1-13&keywords=inateck+usb+3.0