There are a number of variables you need to consider in order to space them correctly.
First, how much light do you want? Brighter rooms require more lights. Related to this is the max wattage of your cans. You will need more 35W lights to achieve the same brightness you'd get with 50W cans.
What type of spread (angle of light ) do the bulbs you plan on using produce? Bulbs with a wider spread will cover a larger area but light will be more diffused.
The typical rule-of-thumb is 24" from each wall, and then 3-5 feet between cans. So in a 12' room you'd probably want two rows of lights. Considering it is a bedroom and you don't need it as bright as a kitchen, you can probably get away with 2 rows of 4 lights for a total of 8. If you are using 50W bulbs then you will have 400W of light in your room.
The height of your ceiling also plays a big role. Higher ceilings typically require more cans since the light diffuses more before it hits the floor.
Also take into consideration the position of ceiling joists; this might limit where and how many you can place.
I always recommend drawing out your design on graphing paper before you make your first cut. Find out the spread of your light (taking into account the ceiling height), and draw a circle under each can with a protractor. You want the light produced on the floor for each can to slightly overlap another light so that thre are no dead spots.
Make sure you check the ceiling joists before you start cutting!
I should also mention that if you are installing into an insulated area, make sure you use IC (insulation contact) rated cans!
Best Answer
They need replacing
From the sounds of things, these are LED recessed fixtures with integral drivers. Unfortunately, that means that if a power surge kills them, its off to get them replaced, unless somebody around's up for light fixture surgery that is. The good news is that it's not terribly hard to find replacement fixtures as recessed lights have settled on a few "de facto" standard sizes for some time now.