TPS61169 LED driver not functioning properly
With a total load of 9 watts, given that the efficiency of the boost circuit might be 80%, the input power might be about 11 watts. So that would be a continuous current of 2.2 amps. However, the boost circuit takes energy in bursts; possibly 50% of the time it extracts energy from the supply and then hands this energy to the load for the other 50% of the time.

So the average power taken from the supply might be 11 watts but in reality it will be something like 22 watts for 50% of the time and zero watts for the other 50% of the time.

So, the peak demand is 22 watts at 5 volts. This is a current of over 4 amps and your power supply cannot hack it. Try just one LED to see if this works. Also double check that the peak current in the inductor is what you think it is. I have my doubts.