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It all depends on the level of your "relations" with them. The higher they are, the lower a fee they will ask. I dont know the exact values, but off the top of my head, it goes somewhat like " 90 or more out of 100, they will "love" or "worship" you at that point, and if you ask them to join you, they will join you for free. If it somewhere between 75 to 89 out of 100, they will "like" or "love" you but will still demand some money to join you are a servant. It is somewhere between 60 to 75 out of 100, they will "like" you and will demand a lot of money to join as a servant. If its any below that, they will most likely decline your offer.
Most prisoners start from "loathe" you which is somewhere between 0 to 15 out of 100 relations. Talking to the prisoner in court to ask "how they are doing" should raise their relations a little bit - but you can only do this once a day unless you "reset daily interactions" by use of some items, like Cider, for example. You can also raise "relations" by giving them specific items, such as "trinket" as one example, or by feeding them before they feed automatically when their nutrition is lowered.
Those who have "warden" perk will gain relations with prisoners faster. You can also raise relations passively - for this, assign the prisoner to be imprisoned in the "treasure room" with the Keeper character, and make sure that you set them to "socialize". This way, Keeper will talk to them on occasion which should over time increase their Relations.

BUT

By far the easiest way to raise a prisoners "relations" is by having lewd relations with them, and pushing them to the "horny" temp perk - this buff has a 30% chance to increase "relations" by 1 every tick. The thing is, since prisoners are prisoners, they cant, in fact, refuse when the Keeper propositions them to have sex. Servants will only agree to have sex (perform lewd actions) based on how much they like the keeper and how horny they are, and "asking to get lewd" is a daily action too, so you can only ask once. But, with prisoners, they will not refuse, In fact, some "servants" can start from "loathe" relations too, at which points it would take days for them to become friendly through normal interactions, and those would refuse to have sex even if significantly horny, and during these days they will often get into fights and shouting matches with other servants, causing trouble.
Easiest solution is to select such a troublesome servant, go to "actions", choose "emprison" in the "actions" in dungeon management screen (I assume typo, meaning "imprison") and select a jail to jail them in. This turns a servant into a prisoner, and reset their "relations" to 0 ("loathe you"). Then, select yes for them for enter court and go into court. In here, they are bound, they "loathe" you and you can force them into sex with 100% success rate, although they will complain and cry and so on (which makes sense, they are being forced upon). If you want to make them compliant, then I recommend selecting the option to "temporarily release" them, and then select "follow me" - this gives them temporary massive boost to "relations" - it will catapult you from "loathe" all the way to "like". Since they are still prisoners, they still cant refuse your sexual advances. So, have them agree to "get lewd" and select your preferred activity. Now let the keeper and prisoner perform it over and over and over again till the prisoner feels like "worshipping" you - at that point, you can cease to have sex and select conversation option "will you serve me" to which they will gleefully agree. If your court has other servants inside and props like toilet/shower, you might wanna check those props first and harvest all the "bottled" resources first. If you want to go AFK while they perform the acts, I recommend selecting "blowjob" option as that one will not result in pregnancy (unless that is your wish, of course).

So, to sum it up: If you have prisoner/bad servant you want to turn into a good, loving servant, first imprison them, enable them for court, go to court, temporarily release them, ask them to follow you, have sex with them and go AFK for some 5-15 mins or so while they constantly perform the naughty, at which point the prisoner will worship you, stop the sex and ask them in conversation option to "serve you" - court automatically ends and you have a new servant who adores & worships you. This is the most efficient brain washing method in the game that I found so far. Other variations of this method includes the use of props like "milking station" or "lust vaporizer" which will increase the lust of the prisoner gradually if used in the court, so you can make em horny without touching them yourself. Naturally these methods can be troublesome and require more effort, but they are still an option.