How to permanently curse an existing item?

The idea of item that can't be removed is against concept of most spells and effects and items in D&D 5. So, if you want an item that "can't be removed by remove curse" - it's close to impossible and whatever such item would do - it likely would belong to area of 'legendary' things, some of which break the normal rules. And certainly, would be a bit too overkill for a prank.

But, technically speaking - there are several ways.

First - there is a Wish spell. 9th level, alters reality. Probably it can alter the king himself to make him stutter or that he is an idiot each time, when he says his name. You need to be at least 17th level to do it (on par with level requirements for creating legendary items) and willing to face consequences of wasting potentially game world changing spell on something that trivial.

Second - you can find a friendly god. Or at least friendly cleric of a god, who would be sympathetic to your cause. (Probably some trickster god, if he won't decide that king did right thing by deceiving you, or some god of law, if king had done it unlawfully). And your cleric friend needs to be at least level 10 to be able to pray for Divine Intervention (level 20 for instant response, otherwise you may spend several days or weeks paraying) - it's a bit random and it's not guaranteed that you'd achieve the desired effect - but it depends on creativity of your DM.

Third - while it's not exactly creating a cursed item - to achieve your desired effect you can use geas. 5th level spell that forces target to obey your command. It's not permanent - 60 days if cast with level 5, a year with 7th or 8th, so you may need a way to recast it every now and then and target may pass the initial save or just endure 5d10 damage each day. Also, you need to somehow hold king within 60 feet from you for a minute as you cast the spell and make sure that he won't remember/identify you, unless you want to end up as 'most wanted people in all surrounding lands'.

Fourth - work with DM, talk with DM, discuss your desires with DM. All of those ways require it anyway and his input on whether god would interdict on your behalf, whether Wish spell may have such effect and how you can stealthily put Geas on the king. If your DM is creative - he may come up with some way to do it, to craft a special spell, perform some ritual and so on. If your DM is against such prank - he would find a way to stop you from doing it even with a Wish and, potentially, it may lead to ruining the game for both sides. Honestly, it's one of the worst things that may happen with game - forcing DM to do something that he doesn't want to do.