I can attest that memorizing a chapter of the Bible is not as hard as it sounds. It's not one big mammoth undertaking. The process is actually quite familiar, just spread out over many days.
Start by memorizing the first verse. This step is literally no different than when one is memorizing a standalone Bible verse. How ever much one learns, call it a day.
Every day after that has a few steps:
1. Review what was learned previously.
Recite as much as you know from a cold start with no previous review that day. This will tell you how effective previous efforts were. If one makes it all the way through perfectly, then it's on to the next step. If not, it's good to review what one has already learn to get it down solid.
2. Learn another verse.
Given that this starts with the reading step, it may help the stitching together process to include the end part of what one learned previously, and practice that together with the new verses one is learning.
3. Stitch it together.
There are two parts to memorizing a chapter of the Bible that may be unexpected. What can feel like an unexpected surprise, especially in this context, is that the verses already learned and reviewed on previous days, one still knows and gets better at each day. The unexpected challenge after learning a new verse is that parts of the old and parts of the new can both feel unfamiliar when they are put together at first. This is normal, and it just takes some patience and practice to put the old and the new together. Average time: 5 minutes at most.
4. Reinforce it.
Recite what one has learned of the chapter three times in four directions, at the walls of a room, for instance. If what I wrote on this previously seemed overdone, it is all highly relevant at this stage, especially the further one gets through a chapter. Average time: 15 minutes, for up to half a chapter.
I have made the most progress when leaving myself an hour in the morning for just memorizing and meditating on what I've learned. When not that much time is available, a half-hour is also useful for learning a verse, stitching it together, and reinforcing it.
If I am consistent and making progress each day, I can memorize an average chapter (25-30) verses in two weeks. That's going at a pace of memorizing about two verses a day which is a very attainable pace.