Fixing an LED night light circuit
Each White 5mm LED is rated about 65mW at 3.1V@20mA with an ESR of approx 15 Ohms intersecting with 2.8V.

Thus 8 parallel LEDs have an ESR of 15/8~=2 ohms. The fixed part appears have gold indicating a decimal point between Red-Red or 2.2 Ohms .

Thus current at 4.5V to 2.9V to a load defined as 2.8V +(15/8+2.2)If=Vbat

Thus If =( Vbat-2.8V ) /  4 Ohms 

For Vbat =4.5 , If= 53mA / LED. !!
For Vbat =4.0 , If = 38mA / LED !
For Vbat = 3.5, If = 22mA / LED ok.
For Vbat = 3.1, If =  2.8 mA/ LED dim.  

So power consumption for average at 4V is 1.2 Watts and 3 cell’s with est 3* 1.8Wh @ 0.5A  might give 4 hours really too bright then a few days dim to dead. 

Conclusion : 

bad match of battery to load.  Increasing R to two 2.2 R’s to share heat or 4.4 reduces power drain, reduces efficiency  but reduces the initial current giving slightly more time.  Maybe 2 weeks vs 1.

Chalk it up to a poor design.