I don't know unity, but I'm sure you have some kinds of variables stored somewhere you can access. The easiest thing to do might be to have two variables that are each arrays of Vector2s. So, you would have one that is like "bedroomPositions" and another that is "kitchenPositions." So, each time a room is loaded, all the items can be put back into their places. Here is some pseudo code of what that might look like
//For each item when placed
if item.type == "kitchen"
{
kitchenPositions[item.id].x= item.x_position
kitchenPositions[item.id].y = item.y_position
}
//when loading a room
for item in kitchenItems
{
item.x = kitchenPositions[item.id].x
item.y = kitchenPositions[item.id].y
}