Given your existing parts stash, above is what I deem to be the closest-exit.
The external components are two (2) 2N3904 transistors which you have...and one (1) 47uF 16V capacitor. These capacitors should be 5-cent parts but of course that's not the way the world works for the DIY hobbyist! Note that an external resistor is NOT needed.
I found a capacitor assortment on both Amazon and eBay which apparently ships from within US for about $5 so about 5 cents per capacitor. They appear to be the same assortment.
I tried to figure a method to modify the module itself but gets fairly tricky with tiny surface-mount parts.
I realize the external modifications sort of demotes the low-cost module concept. And perhaps there is a true Delay-ON module out there for $1 that works out-of-the-box so to speak. Note that the same external modification will work for 5V or 12V DC power ... of course you must attach an LED strip of the corresponding voltage!
