Just building a nearly airtight castle heated by steam will go a long way and you don't need modern materials though that helps.
Even if the materials themselves don't insulate terribly well, if the structure can made as airtight as possible, the energy required to maintain a given temperature will go way down. Sealing cracks with pitch or tar should help a lot.
Stone and wood are naturally decent insulators, not as good as fiberglass insulation but plenty good.
Older homes before central heating and air conditioning heated themselves with steam. Water was heated in a boiler which was then fed into a system of pipes and radiators. Middle Ages blacksmiths certainly had access to cast iron technology so making the boiler, pipes and radiators shouldn't take too much experimentation.
Inventing threading will be a bit tricky but if backed by the King, it shouldn't take too long to describe to a competent blacksmith how to do threads. (Edit: Okay, threads are crazy crazy tricky with high precision and metallurgy requirements. Perhaps all you could do is get them started.) The architect may have to invent measurements to standardize on but that's not too hard.
Safety testing will be essential because it won't do to have a radiator explode in his majesty's privy.
Design of the castle
The architect will have taken an architecture history course so he need only examine the king's army to decide how to build a maximally defensive castle. Or, if the architect is really lucky, the king will have him design a castle without all the required defensive measures. The design of defensive structures changed with advent of firearms and cannon so the architect will need to design appropriately.
Anything, absolutely anything that the architect comes up with will blow the king's mind. It will be a structure hundreds of years before its time. A good architect will get to know his client's sense of style and make something astounding and approachable because it will be very easy to design something that offends the king's sense of style. However, if the architect does well, everyone in the known world will want one of his castles.
Glass is the middle ages was super duper expensive so making double walled windows will be exorbitant by any standard. It's up to the king to decide on that one. It's a big bonus if he can pull it off. Every piece of glass was cut from a giant clear wheel of glass. Very time consuming to make.
If the architect is familiar with the float glass method of glass making then he could suggest it to the blacksmiths/artisans of the king to get large quantities of high quality and perfectly smooth glass for windows, mirrors, whatever.
Removing ways for mice or rats to invade will be huge. Disease goes down, food spoilage goes down.