Recommendations for Accommodations in Split

This is where I stayed a few years ago, it was very nice (definitely not a cheap option though). I think it's totally worth comparing prices direct vs. booking.com or the like. You may be surprised.
http://palacesuites-split.com

I also stayed in a simpler, less expensive place with a friend. It was perfectly fine, and same central location as the other one. Look under "Special Offers" at the top for further discounts.
https://www.kastelsplit.com/en/split-hotels-private/

One recommendation I can make, since I learned the hard way, is that there's no need to take an expensive private transfer from the airport. The Croatia Airlines Shuttle bus drops off close enough and a very fine, inexpensive alternative. The center of Split does not allow cars anyway, including taxis and private transfers. They park as close as they can, and then it's walking and rolling with your luggage. My private transfer was booked with the hotel so, of course, he was incredibly generous in carrying everything all the way up the room.

In summer 2018 I stayed for several days in Rooms Toni Palace. Fantastic location, in the Diocletian Palace area, large lovely room. Good (tho cold) breakfast. However, I cannot find on-line how to book them directly. I used booking.com (as I do for most of my European bookings) and all went quite smoothly.

Last August we stayed at the Hotel Peristil also located within the walls of the Diocletian’s palace. The location was fantastic, not too expensive, and they had a wonderful hot breakfast included. The owners were very friendly. We also booked through booking.com with no problems.

Linda, I can recommend Hotel Globo. We stayed there last September on our RS Adriatic Tour. It was comfortable and quiet location. We had a wonderful dinner (Mediterranean) at a lovely neighborhood restaurant. They have outdoor dining in a gorgeous setting. The restaurant's architecture is from the art nouveau movement. It was a short 5 minute walk from the hotel. www.restoran-perivoj.com. Enjoy your stay!

I enjoyed a place recommended in RS Croatia/Slovenia book called Hotel Villa Diana. We loved the 3rd floor room but know that there is no elevator, also stayed in a very nice 2nd floor room (outside the walls by 5 min walk) with parking if you have a car. The Villa has breakfast and when I was there had a restaurant serving dinner ... authentic neighborhood; Tommy’s Market practically across the street with deli meats and fabulous selection of cheeses), good small restaurants.
(2). I have stayed on the north side of town in a small apartment in an older estate just minutes walk from Ivan Mestrovic Gallery (as noted in the USA for La Pieta sculpture at Notre Dame University and the two equestrian mounted Indians near Michigan Avenue, Chicago). The access to the water from Apartment Lju at the Beach and strolling the very private promenade on that side of Split makes this accommodation unique (parking available).
(3) Lastly, mI have stayed fairly close to the ferry/fast cat port at Rooms Anna - sweet upstairs apartment and very very reasonably priced. Owner is lovely and helpful. Met me at ferry to help walk the shortest way to the apartment (at 7am... I had booked the room including the night before so that I had immediate use of room that morning). Just 100 steps to a recommended RS guide restaurant formerly Black Cat, now called Toć - breakfast, good Mexican food and margaritas, good Croatian typical cuisine... the Konoba just down the street is excellent.
Villa Diana is also in this neighborhood.

I'm not sure what you are looking for in a hotel. If a really nice hotel with large pools then you might want to check out Hotel President in Solin which is a few miles from Split. We stayed there on an Insight Vacation tour in 2011 and it was nice. It's spacious with nice rooms and not like the typical backdoor hotels that are recommended by Rick Steves. Its close to the small town of Solin but not much else is within walking distance.