Here you have the manual for the computer, Lenovo H50:
http://www.produktinfo.conrad.com/datenblaetter/1200000-1299999/001283470-an-01-en-HARDWARE_LENOVO_H50_50_90B6000AGE_PC_SYS.pdf
On page 26 and subsequent (as per the footer page numbers at the bottom of the page) you have a sketch of the motherboard (there are different models). You have to locate the Front Panel connectors, which include the Power Button. They are at the bottom right corner of the mobo. In the manual, there are no details on which pins of these Front Panel headers are for the Power Button, but if you look closely on the mobo itself, right around the header, there should be codes (maybe something like PWRBN for PoWeR ButtoN, or something like that) written indicating which pins are for what. Note that some of these cables have polarity (i.e. if they don't work when plugged, try them the other way around, as maybe you hadn't correctly aligned the + and - poles of the cables).
If you manage to do this, you won't have to buy a new mobo. But if you have to, you can probably find some on ebay or some recycling store (there are many on ebay that recycle old computers and sell at very competitive prices the working pices for parts).
Hope it helps!