Like PP say, you appear to have colour build up on the ends. This needs to go before you think about anything else.
It's way too early for you to "embrace the grey". You only 20% (ish) white and this is probably more crown and temples and less at the back. Most women do not turn "grey" (no such colour) becuase the white streaks are uniformly distributed, giving an overall impression of your own colour but lighter. White streaks and patches on dark brown hair is not a good look, especially if you have to wait at least a decade, probably closer to two decades before you go completely white.
Your problem is the contrast between natural colour and white, which is like 7-8 shades. This means any white root regrowth will be obvious and you need constant maintenance. Your other problem is that your white hairs appear to be resistant if your box dyes wear off.
Given that you have a long time to wait for fully white hair, which you can then deal with as one entity, you have to find a long term solution which deals with the dark brown AND the white.
- You can simply dye the whole lot your own base colour, which looks to be a 4 (1 = white, 9 = black). You can buy professional colour and peroxide developer. You will probably need to use 20 vol (6%) peroxide. This strength of peroxide will open up the cuticles enough to ensure your resistant whites grab the colour, so it fades slowly or grows out. You need to learn to do the roots only by doing it in sections or getting a friend to do it. If you do not apply only to roots, your ends will go black due to colour build up.
You will probably find that the colour takes slightly differently on the whites, giving you a nice lowlighted look, which will add contrast and avoid that block colour look.
- As above, but gradually, you can start mixing colours, so every 6 months to a year you might want to make your mixture lighter. This will accommodate the increase in white hairs. So I'm thinking change the ratio like 4:1 base 4: base 5. This slow and gradual introduction of a lighter colour will mean that as you get whiter and whiter, the contrast between existing dyed colour and ever-whitening roots becomes less stark. So you will end up looking more like a base 5-6. Mid to light brown. This avoid the badger look between root touch ups.
Eventually the whole lot will be white and you can do what you like with it then. Dye it blonde, red, add highlights lowlights.
- Do as PP have suggested, start blending in the whites with a highlights. A nice caramel colour would look great on your hair. The re-growth will look less obvious and you can ask the hairdresser to increase the number of highlights as your hair gets whiter as the years progress. You've also got a few more options for variety here, as a professional will be able to change the colours, so you can have different shades and tones. This would be my preferred option if you have enough money and time. I'm thinking £100 per month depending on where you live. Also lengthy appointments. If you ever run out of cash and time and stick on a box dye from superdrug, you'll ruin the effect and have to start again, so this needs a long-term commitment of time and money.
There are lots of professional colours you can buy online of from your nearest branch of salon services. The assistants are often professionally trained and will advice you. There also have shade charts. My last piece of advice is to do a strand test of your chosen colour; they often turn out much darker than the shade chart and you may need to get a higher number like a 6 in order to achieve a 5