The most popular story - from a filmmaking point of view at least - is that of Elijah. Technically, most of Elijah's story falls within 1 Kings, but volume 2 does contain the closing scenes from his ministry, most notably him being taken up to heaven. But films about Elijah are rare: I'm only aware of three which cover events in 2 Kings and only one of those is a feature length film.
Technically, the first of these isn't even really a film about Elijah, rather it's about his nemesis Jezebel. I've never actually seen Sins of Jezebel (1953), but seems to cover most of the events ascribed to her in the Bible, from her appearance as Ahab's wife in 1 Kings 16, through to her death at the hand of Jehu in 2 Kings 9. I suspect it takes the principles of the sex, sword and sandal epics to an extreme, but one of the commentators at IMDb describes it as a "fairly straightforward retelling of the Bible story". Production values, however are reported to be low, but then it was filmed in 3 days on only a $100,000 dollar budget.By far the best that I've seen is the Elijah entry in the Testament: Bible in Animation series. It's hand drawn, but highly stylised, animation accompanied by operatic sound courtesy of Bryn Terfel, and the BBC National Orchestra / Chorus of Wales. I've discussed this film before, but I'm struck on this occasion of how Elijah reminds me of a younger Brian Blessed.
The other Elijah film, which I have also discussed before is Elijah, a Fearless Prophet from the Living Bible series. Both these Elijah films include his dramatic ascension to heaven, but omit the other main story from 2 Kings where Elijah condemns Ahaziah. I also yet to see Elijah and the Widow of Zarephath (1996), but I suspect from what I've read that it doesn't go as far as 2 Kings. (Neither does the Elijah excerpt from Friends and Heroes FWIW)
Elijah's servant and successor Elishah has faired even worse. Whilst he does get a bit part on those Elijah films, there's only two "films" where he plays a more leading role. The first is The Son of the Shunamite from way back in 1911. It's described as a classic horror film, where Elisha raises the widow's son.There's also a more recent cartoon which covers the story of Elisha and Naaman. Riding for a Fall is part of the Bugtime Adventures series. As far as I'm aware it's the only version of Naaman's story. Quite where the bugs fit in is, at present, beyond me.
However, it's the middle part of Kings where there is a real paucity of film coverage. As far as I'm aware there isn't a single film which so much as touches on anything between the end of Elisha's ministry and the beginning of the events leading up to the fall of Jerusalem. Because that incident was so significant it spills over into other stories, particular those of Jeremiah and Daniel, and so there are a couple of films which deal with these events.Slaves of Babylon was released in 1953, the same year as Sins of Jezebel, which, given the general shortage of films touching on 2 Kings, is quite surprising. It starred Richard Conte and Linda Christian as two fictional characters Nahum and Princess Panthea. The most well known 'biblical' character is Daniel who doesn't actually feature in 2 Kings. I now realise that I missed this film out of my recent look at films about Daniel. For those who are interested there are a good number of photos from this film available here with some original pictures and a poster also available on eBay at the moment.
Finally there's Jeremiah from The Bible Collection which is where I got the heading image (and the one I ended up using). I've been an admirer of this film ever since I first saw it. Whilst the Bible Collection's insistence on bringing a romance element into every story they handle is present here as well, it's less of a distraction, and doesn't detract from the fine work by Patrick Dempsey in the lead role. It's main achievement is highlighting sufficient narrative to make an interesting plot, whilst including enough prophecy to make the exercise as a whole worthwhile. In contrast to Slaves of Babylon, however, the fall of Jerusalem occurs at the end of the film rather than at the start.