What you have there is the dreaded BenQ-designed CCFL inverter design that permeated all 17-19" TFTs in the mid-00's. You will likely find many more of the exact same failures in almost all brands of CCFL-backlit displays of that era, because with very few exceptions all commodity displays used the same CCFL inverter design. The biggest problem with this design is that BenQ designed in freestanding 2SC5707 transistors for the (bipolar) power switching, which are designed for 9-W (peak) 15" display backlights. Then they reused the design for 17 and 19 inch displays at higher brightnesses, driving the transistors beyond their design limits. This caused them to overheat (you will see a bit of scorching near those transistors) and eventually fail.
The problem was compounded by a bad default choice of capacitors, which are often faulty as well. These capacitors would get higher ESR which caused the primary power stage to ripple more and exaggerate the overloading of the 2SC5707 transistors. These problems are technically unrelated, but one often leads to the other.
I can't really read your brand and type of capacitor from the photos, but at first glance they look like quality brand and my experience with Dell monitors is that they usually replace the default BenQ choice with better caps. That means that the repair is probably as easy as getting a couple 2SC5707's from eBay and replacing them on the board. The fundamental problem still persists though, and it is advisable to improve the heat dissipation capacity of the transistors by e.g. attaching a bit of copper clad board on the flange.