The kids are too small in the frame, diminishing their importance. Also, if this is a photo for a Christmas card, you should get rid of the pumpkins and fall decorations on the mantle. It's the details and the attention you pay to the background and composition that makes the difference between a snapshot and a professional-looking image. The kids are too close to the background. Move them forward and get in closer to them so the background won't be too sharp and won't get shadows from flash (and in the case of the glass in the background, it will reduce distracting glare). A fireplace with no fire can look boring. Consider dressing the kids appropriately for winter (e.g. sweaters). If you're going for the "Chrismas morning or Christmas Eve in pajamas" look, then maybe have wrapped gifts or the Christmas tree in the frame (but you should still groom them so they look good). When photographing children, it's helps to have two people: 1) the photographer, and 2) the child wrangler. If you absolutely can't get someone else to help, then you need to move in closer; the farther away you are from the children, the more difficult it is to control them (this is only true for posed photographs; if you're looking for candid photojournalistic photography, then it helps to use a long lens and keep your distance, letting the kids be themselves). I find that confining the kids to an area helps keep them in place. Some times this is as simple as putting them on a pedestal, footstool, chair, or a makeshift stage (if your fireplace hearth were elevated it might have helped). Kids lose interest quickly, so try to have everything set up before you put them in place. Use dolls or something to test your camera settings and lighting before you start photographing the kids. If the kids get bored or anxious, take a break. Try photographing the kids at a time of day when they're happiest.
As for what settings to use, it really depends on the light, so I can't give you a definite aperture and shutter speed combination, but I can make suggestions. Ordinarily I recommend fast shutter speeds when photographing kids, because they can move around so quickly. However, in this case, you want to use a moderately slow shutter speed so the warm, ambient light, string lights, and fireplace can develop. Slow shutter speeds require a steady camera, so consider a tripod. As long as you drag the shutter, it's okay to use flash so that the kids are nicely exposed. Light drops off quickly, so if you dial down the flash and get close to the kids (and get the kids farther from the background), the flash will help expose the kids, but won't expose the background too much, so the slow shutter speed can let the string lights will develop. Use second curtain flash, because people often think the picture is over as soon as the flash goes off. With on-camera flash I normally recommend bouncing it slightly above and behind you or to the side to get directional soft light. However, your SB-400 flash only tilts up and down, not sideways. Even with cathedral ceilings you should be able to bounce flash if the ceiling is light-colored. When photographing indoors, light bounces not only off of the ceiling, but it also bounces around the walls. I recommend taping a sheet of paper to your flash head, so it softens the light going forward. and creates catch lights in the kids' eyes. Make a fan pattern, so it's widesg at the top and tapers down where it attaches to the flash head.
As for the aperture, if you can keep the kids' heads on the same focal plane (the same distance from the camera), then you can get away with using a wider aperture. An aperture of f/5.6 should allow enough wiggle room so that if the kids move slightly their eyes will still be in reasonable focus (always focus on their eyes). An aperture of f/8 will give you more wiggle room, but you really need to make sure that they're not too close to the background, so the focus is on them rather than the background. It's okay for the background to go dark, as long as the kids are nicely exposed and the string lights in the background is exposed.
When combining flash with ambient light, you control the ambient exposure with the shutter speed and you control the flash exposure with the aperture and flash compensation. Shoot in manual mode.
If you need to increase ISO, try not to go above 800. If you can get the kids to stay in place, use a tripod. Using a remote shutter release can help you interact with the kids when taking the picture.