Firstly unfortunately damaged hair can never be permanently repaired, all you can do is patch up the damage temporarily. The more damaged the hair the more work you will have to put in, instant gratification in most commercial products comes from coating hair with silicones to give faux shine and slip not from healthier hair. Be sure to trim off any split ends or mid shaft breaks, these can catch on healthier hair higher up damaging that.
Choose your products based on your hair properties and the ingredients, best for damaged hair are coconut oil, hydrolysed protein, ceramides and panthenol. The published research on the lauric acid in coconut oil penetrating, reducing porosity and protein loss (further damage) is overnight to twenty four hours on dry hair. It will absorb more quickly with some heat but for badly damaged hair an hour may not be enough. Honey is a lovely combination with coconut oil, it is a powerful humectant so helps moisturise.
Egg proteins are too large to penetrate, if you want a totally natural source of hydrolysed proteins that are small enough to stick or absorb use gelatin. There is a recipe on the DIY boards, or I rate the Komaza Care products (protein strengthener is hardcore so can be somewhat drying, Matani repair treatment spray is gentler, but also contains ceramides and panthenol). You generally need to deep condition for moisture after a hardcore protein treatment.
Many oils and butters like shea and sunflower don't really penetrate, they are best used in sealant products rather than deep conditioners. Argan and avocado oils are rich in oleic acid which can penetrate ... if there is a decent amount in the product, they are pricey ingredients so generally there is not. They also may not penetrate in any worthwhile amount in the time we usually deep condition for.
If you do want a ready made deep conditioner I quite like Crown Pride tucuma butter mask (rich in lauric and oleic acids, honey) which is super thick so works best with heat, or Hairveda Sitrinillah (coconut oil, castor oil, glycerin) which is more 'slippy' like a regular deep conditioner.