My guess would be that you should be able to use the laundry water directly for gardening purpose. There are no toxic compounds expected, except for some detergents (which if not too high concentration would be broken down in the soil), while also organic matter will be quite low. P and some N might be welcome additions for plant growth. If detergent concentration is high you might need to dilute with fresh water when watering the plants. Detergent ingredients include a.o. surfactants like alkylbenzene sulfonic acid, builders which have sodium nitrilotriacetate, sodium triphosphate, and zeolite, and bleach-active compounds. Good luck with your trials.
Information about the component LAS (linear alkylbenzene sulphonate) that is present as cleaning and foaming agent in all household detergents, but also biochemical decomposition pathways are detailed in the attached document. You will see that there are some environmental risks since not all compounds are biodegradable but dilution could be a solution. Also, too much Nitrogen and Phosphorus (coming from nitrit and phosphate in laungry waters) discharged on a limited surface has a negative impact in terms of eutrophication. You can find details about alkylbenzene sulphonates break down (via the intermediates sulpho phenyl carboxylates SPSs that is tens times less harmful than LAS) and also about the parameters that are considered in assessing the laundry waters quality.
I hope this document will be helpful for your project.
Dr. Farid Benyahia has nicely described the characteristics of waters from Commercial laundry (traditional coin operated) and Industrial Laundry (services to hotels, large companies, etc..).
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I think there are two key points in laundry water. One is detergent, the other is microorganism. It needs adsorption and disinfection. Adsorption would reduce detergents, organic matters such as personal care products, and dyes presented in the wastewater.