Iphone 5 is refusing to turn on
If an iPhone or iOS device doesn’t self start after 15 minutes of charging, the proper step is to force a reset and then repeat this pattern four or more times before seeking a repair of the hardware. 

Since your home button is broken, you could try disconnecting from power and reconnecting at each 15 minute point and hope. 

You likely have two problems that need fixing, though. The button and the power circuitry or the electronics to start / run are failing. The tech (or you) will need to open the device to figure out which.