Heading into a major renovation without a detailed plan, materials schedule and a clear budget can turn what should be a gratifying experience into a nightmare. Preplanning is the key to success. Take your time, if you don't have the ability to create detailed drawings of the new layout, then it may be wise to have an architect create some plans for you. There are lots of items to be considered, plumbing, electrical, structural changes etc.
You should spend time at home centers or specialty showrooms so you can determine what kind of fixtures, cabinets, counter tops, flooring etc, that meet your needs, style and budget. Remember, quality of these items varies tremendously, so do the costs, so get a handle on those items early in the process.
Create a "schedule" or list of the exact items you want in your renovation. The more detail, model numbers, colors, accessories, the better. Even have samples of the flooring, trim, finishes etc. Many places will loan you a sample cabinet door and small samples of counter tops.
The advantage of having an architect is that he/she can take your selections and work them along with the unseen requirements into a complete "Scope of Work" that can be used to solicit quotes from contractors. The architect may also help you in constructing a request for bid that will include items such as time schedules, schedule of payments, and final hold-back (10% is common) until a final acceptance inspection is completed. The business aspect of a project like yours is very important since you are probably talking $30,000 to $40,000 or more.
When selecting the contractor, vet them thoroughly, check references, go look at some of their recent work and talk to their customers. Don't be shy about it, I have seen and cleaned up after too many botched and unfinished jobs because the customer fell for a smooth talker and never checked them out professionally. Be very cautious.
I could probably write a book based on your question, but I hope I have covered a few of the bases. Feel free to ask follow up questions or more specific questions. Good luck.
You can definitely get flexible led strips that are bright enough to do what you want. Many of them run on 12V, which you can easily get from a small power supply, and you could just plug it into an outlet on the counter.
I would put them under the cabinets; it would be easily to hide them there. Many of the strips can be had with adhesive backing, which would just stick under the counter.
To answer your question on the individual lights, the individual LEDs will blend together nicely.
If you look at Amazon, you'll find a lot of solutions, and they are fairly cheap.
Best Answer
Although code is not a limiting factor the light density will be. Based on homes I have built and redecorated I would have under cabinet lighting at the stove top, and a light above at each end shining down on the fridge and oven. Then 2 or 3 in-line at the island. The actual spacing depends on the fixture themselves.
Some recessed lights have a narrow beam some are quite wide. I tend to split the width of the oven and fridge in this case and mount the fixture centered and 1/2 the with of the opening away from the cabinets. With this method any openings at the bottom or top have light. If two close the bottom shelves are dark, and the cabinets reflect the light away from where it is needed. Two far away the upper shelves are dark and you have shadows when standing there looking.