Check if the board is in sleep mode I'm using seep sleep mode with UNO with a simple sketch that turning on an LED and then goes to sleep. as far as the ATmega328p datasheet in sleep mode part showing the wake up sources and deactivated clock sources in deep sleep mode and the I/O clock becomes deactivate so we expect when the board goes to sleep the LED should become off because of no clock. but it doesn't happen and LED keeps ON in sleep mode. what is the problem?7 and how can we even find out that the board is on sleep mode or not?
The IO doesn't require the clock to keep its current state. It only requires it to change state.

There is no real external signal provided by the chip to indicate if its in sleep mode or not. However you could monitor the current consumption of the chip and when it drops below a certain threshold you can assume that it is in sleep mode.