As others have said, it isn't as good as meta-junglers, but it is certainly something that can work, and could even be good. Really anyone CAN jungle, but the biggest hurdle for any jungler is their first, and sometimes even their second full clear. I think absolute worst champion you could ever attempt to jungle with would be Azir, but if you could struggle through the first part of the game, he can clear jungle camps really fast.
As you said for Leona, she has a lot of hard cc, which is great for ganking.
Her decent base health and bonus resistances help with her ability to mitigate damage while clearing. Her auto attack reset and AOE damage really helps with her ability to clear with decent speed, but not fast. I'm not sure about her mana sustain, but the Blue buff and jungle item will really help with this matter. Sheen / Iceborn Gauntlet would nearly solve any mana issues all together.
I don't think full damage is really the best way to go about it though. As I said, her clear speeds are not ideal. Building more damage will be important for her to keep up with her team and to have an impact when she engages.
Her auto attack reset actually makes Tiamat / Titanic Hydra pretty valuable, despite the fact that she doesn't scale on AD, but it does have good synergy with Cinderhulk. I don't know that Stalker's Blade would be 'better' than Skirmisher's Sabre. You're already layering a lot of crowd control from her kit and with Iceborn Gauntlet. The Challenging Smite will actually work to her advantage because it gives her 20% damage reduction, and increases her damage output with a true damage burn.
Personally, I would decide on the rest of my build situationally. If you're ahead, go for more of a bruiser build so you can help your team snowball. If you are behind, you will need to be able to help your team defend and teamfight around objectives.