Like Elainera has said they had some issues early on which began the delays. I have worked on drilling rigs years ago and I remember most of the steps.
The first step they had to do is regular land escavating to flatten out the area around the hole so they can bring in the large drilling rig which is a large truck that does the drilling. That truck can turn the drill pipe column and allows changing of each of the sections of the drill pipe as they go down. Large oil wells have a fixed platform but in this operation they are using a drilling truck rig.
What a lot of people dont realize is to drill a well that deep, it takes sections of pipe called a drill column and each pipe is usually only about 20 or 30 feet long. So as they go down they have to clamp off the drill column pipe and attach another section of pipe to keep going down. Each section of pipe one at a time gets attached as they go down.
The other thing unique about this 2nd hole they are drilling is the diameter is very large to fit the human cage. The larger diameter hole means they had to use a special drilling bit or method to get a large diameter hole like that. Normally water wells dont have to be nearly that large of diameter like the one the boy is stuck in is rather small diameter.
The other thing that most people dont realize is the drill bit at the very bottom of the drilling pipe column gets dull often. Then what happens is they have to pull up the entire length of drilling pipe back up to the top and remove each section of pipe as they go in order to get to the very bottom one to change out the drill bit. The drill bits are usually commercial diamond tipped drill bits that have rotating edges but even they get dull and have to be changed out if the rock is hard that they are drilling through. Then once the bit is changed then they start all over attaching each section of pipe as they go down.
Once the hole is drilled then they have to put in the entire "casing pipe" which shores up the edges of the hole and keeps any dirt or rocks from falling back into the hole. That casing pipe is also in sections of length due to the length of the hole which is about 300 feet total depth.
Every time they screw on a new section of pipe it takes a little time. The workers can do it fast but still it takes a little time.
All these types of things add up to it taking longer than most people realize.
Here is a video that shows one being dug