RAW: It’ll cost 6000 gp; you should use fabricate
The rules for barding (PHB p. 155) don’t account for size so the armour for any sized creature costs four times and weigh twice that of normal armour (as listed on PHB p. 145). This brings the barding in at 6000 gp and 130 lb. This is ostensibly the cost of having the armour made, although your DM might require the blacksmith to be persuaded to undertake such an odd task.
If you want to save money and have time to spare (and proficiency in Smith’s tools) you could make it yourself. Using the rules for crafting nonmagical items (PHB p. 187) you need materials equal to half the market value (3000 gp) and make 5gp worth of progress per day, a requiring mere 1200 days which is only slightly more than 3 years. Unfortunately, as you’re going to have to get this custom made (presumably) it’s going to take any smith the same amount of work days to get this done (the workdays can be shared between multiple smiths to reduce total time).
If you want a bit more expeditious progress you will need a Wizard friend. (I mean, we all need a Wizard friend.) A 7th-level (or higher) wizard, specifically, and one that is proficient with smith’s tools. The spell we’re after is fabricate which will let the Wizard transform raw materials into armour (among other, similar uses). Fabricate is (for metals) limited to a 5 ft. cube, which depending on how well whale barding can be tetrised into a cube, might require more than one casting (your DM’s call).
Also, if you don’t have a Wizard who would be willing to work for free and have to rely on Wizards-for-hire, that will cost you an amount up to your DM, as there are no rules specified in the Spellcasting services ‘rules’ (PHB p. 159).
The above assumed a whale would need as much (or little) barding as ‘regular’ mounts. These span the size range between Mastiffs (Medium) to Elephants (Huge). While not specifically addressed it is implied that the barding rules apply to all of these, likely in the interest of simplicity. However, your DM might require that effective armour for a whale need to cover more of its body than an elephant needs.
Also, there is no stat block for blue whales (as far I know) and the closest is Sperm Whale (from Icewind Dale: Rime of the Frostmaiden) which are Gargantuan. Blue whales are even bigger, though there’s no greater creature size for them to be.
Therefore, your DM might rule that whale barding is double (or more) that of normal barding, which would apply that same multiplier to all the numbers given above (not the Wizard level). You have 6k gp and 6 years of your life to spend on this, right?
This is of course only the matter of obtaining the armour. Putting it on the whale is also gonna be non-trivial. The rules for getting into armour (PHB p. 146) puts donning heavy armour to 10 minutes. This is probably unrealistic. Whales don’t have handsCitation Needed! and so can’t help itself. Even assuming the whale is cooperative in this endeavour, you are probably gonna have to put this stuff in water unless you want go through the process of beaching and unbeaching (debeaching?) a whale.
Let’s talk about Armoured Whales
We’ll start with the rules-as-written effects of plate armour on a whale. For these purposes we’ll use the stats of a Sperm Whale. Firstly, its AC will become 18 (this was the point presumably). It fulfills the Strength score requirement (26 > 15); however, it is unlikely that whales are naturally proficient in heavy armour, and the downsides to that suck. While losing spellcasting probably isn’t going to affect the whale too badly, disadvantage on any roll (ability check, saving throw, or attack) that uses Strength or Dexterity is quite crippling.
How to train your whale (in heavy armour) is up to you though. I can’t help you. The whale is going to have disadvantage on Dexterity (Stealth) regardless. A Killer Whale has a carrying capacity of 3120 lb, though, so even with a lot of barding, it shouldn’t be encumbered. Quadrupled of normal barding (520 lb) and using the variant encumbrance rules (PHB p. 176), the whale is not encumbered (could carry 520 lb without being so), and even if it were, its speed would only be reduced by 10 ft. There are (to my knowledge) no rules which says being underwater affects such capacities. This seems like the point to mention that D&D is not a physics simulator. There is nothing given in D&D for how armour affects a whale’s ability to swim, but there are a lot of edge cases which aren’t covered super well (read: left to DM fiat).
Wrapping a whale (or pretty much anything else) in metal plating will increase its density. Increasing a whale’s density will likely make it sink. This is going to be very bad for your whale. You have in some sense made a whale-filled metal capsule and might be about to find out whether your DM’s world contains a version of the RSPCA and how aggressive they are.1
Assuming the whale doesn’t drown immediately, there are a couple of long term problems. Firstly, you have made armour out of (medieval) steel and whales live in water. Steel rusts. Armour which rusts is at best going to fall off and at worst we’re back to whale filled capsules — only now with a lovely patina. Of course you could get meteoric iron (which is nickel rich), but that’s gonna be a lot harder to track down.
Secondly, wearing plate armour all the time isn’t going to be very comfortable and your whale is going to have a hard time doffing it on its own. Humans need help putting plate armour on (which is a big part of why knights have squires) and unless your whale has some miraculously dexterous flippers it’s gonna need help.
1: See this list to translate into your local version of this joke.