There are several things that could be wrong but unless you have not mentioned doing something recently to the DVI cable, there is no reason why the DVI cable or connector would go bad. That leaves the LG monitor itself.
You need to indicate whether the LG starts faster if on for 5 or more minutes and then turned off and back on, than from a completely cold start. That seems likely from your description.
If that is the case, I'd bet some electrolytic capacitors in the LG have gone bad. Do a Google search on bad caps in monitors and you will get many hits including methods for repair. It seems some suppliers stole capacitor formulas, screwed up production, and sold many bad caps to electronic equipment suppliers. The caps either don't work or go bad with usage. They will swell and may start to leak.
I had exactly this happen to my Samsung monitor and repaired it by replacing two swollen electrolytics. You will need to decide if you are good enough at tinkering to make the repair or at least check for bad caps.