To help heal her gut, yes, that's the ultimate goal, in hopes of reducing or eliminating food sensitivities and providing a healthier base to hopefully avoid or reduce future Hashimoto's, that she may have some genetic tendency towards. Avoidance of anything that causes the gut to react is very important so that you are keeping it out of an angry, inflammatory, reactive state; keeping it off of the defensive mode so that it has the opportunity to heal. Very often, when allergenic foods are avoided strictly for long periods, the gut becomes much less reactive to them or totally loses its reactivity. Be aware that later re-introductions might work fine the first time or two or few, but that reactivity can re-develop over time. Your good observations will tell you what to do.
Next, there are the things that "we" are all learning now that most of us have been missing as a whole for many decades, such as vitamin D (sun, liver, skins of meats/fish, eggs from outdoor-raised chickens, fats from outdoor animals -- all things we've been avoiding), DHA & EPA (fish oils, organ meats), and iodine (kelp, seafood, non-depleted soil-raised crops -- we've even cut back on the table salt used at home that's had it added for good reason. Commercial foods generally don't use iodized salts). There are vegan ways to get all of these of course, but I'm just presenting what's changed in the modern human diet. Olive oil is another great oil to include in diet. It has a low smoking point which means you can cook with it only at very slow/low temps or you can just use it in a dressing or as a spread. I'm sure you are already avoiding trans fats (partially hydrogenated vegetable oils).
Beyond these, daily probiotic use has been shown to be highly beneficial in keeping the gut healthy and in reducing allergic reactions. I like to pick one that has many different kinds of bacteria in it as they all have a little different action. Those that just say non-dairy often have tiny traces of milk in them as the bugs are fermented on milk or lactose. Those that say "no milk" are the ones you're going to want to go for, for now anyway, to keep the picture as simple as possible.
L-Glutamine (it sounds like but is not at all mono-sodium-glutamate), a natural amino acid, has been shown in many studies to be quite intestinal healing. It's often given to premature babies. Recommended doses go as high as 40 grams for an older child or adult, which is like 10 teaspoons/a lot of stuff, but that's high-end for very sick gut of larger person. Even a fourth of that is a lot tablets to be taking, but I'd try to get 3 or 4 grams per day into your child for a short healing period, if possible, and then try to keep up with one a day. It's quite neutral tasting if you want to crush it into applesauce or add to smoothie or such. I've seen chewable tablets at GNC.
Strong anti-oxidants can be anti-inflammatory and otherwise healing. Resveratrol is one good one, found in peanut butter, grape skins, and grapeseed extract. Bioflavanoids are another (but can't be used in those avoiding citrus), found in skin and white stuff of citrus fruits. They're often found in vitamin C supplements. Vitamin C is a weaker anti-oxidant but a good supplement for the gut anyway. Kiwis have about the highest amount of fruits. Acerola or rose-hips are good in vitamin C supps for those who can't have citrus. Turmeric (curcumin, a curry spice) is one of the most powerful anti-oxidants. It's hard to get a therapeutic dose from diet alone, especially with a child's palate, while I know that the supplements might not be very swallowable now. Cinnamon, all of the Italian herbs, and truly most all herbs and spices except for black pepper are great anti-oxidants and gut healers, when tolerated. Obviously I'm not wanting you to provide big supplement tablets of all of these every day. It gives you lots of options though to get some good stuff in every day, knowing that a young child isn't going to be swallowing a bunch of big capsules. All fruits and vegetables have varying kinds of great healing factors and good fiber that feeds the healthy gut bacteria so those are your major target. White cabbage is one that has some very good studies behind it specifically towards intestinal healing (and may be gas-forming at first so go up slowly in dosage if deciding to use it). It works great in smoothies. Aloe vera is very healing for an angry digestive system and can be found in flavored liquid forms so it's more usable with young children.
Additionally, slippery elm, DG licorice, various mints, chamomile, and marshmallow root are soothing to the digestive system. Isn't it interesting that early candies and treats were made out of so many of these things, maybe even mixed with honey (another healer), while now they're all just corn syrup and artificial flavorings.