There is no flex cord between the button and the motherboard. (See the iFixit repair tutorial for the motherbaord.)
Unfortunately there is none.
Is the battery percentage reasonably high? If it is very low, you should leave the phone at charging for a while and then try to power it up again.
Your issue sound pretty much like a broken power button, however. There are some hints on how to repair it here and some photos here. Many repairshops will solder the button in for about 20-40€.
Don’t hesitate to ask for further advice here. 