Broken tank, homeless newts, and the cycle

I was given a tank with chinese fire bellied newts in a few weeks ago. I didn't have a test kit on hand, so I bought a liquid ammonia test locally and ordered the API test kit online. Ammonia test always came back at 0, so I assumed the tank was already cycled (then newts came with the tank, existing filter and gravel.)

Unfortunately, a week ago the tank cracked and I had to take the newts out. I had a new, larger planted tank set up for them already, however it was still cycling. I thought if I moved the filter and some of the gravel across to the bigger tank immediately, it would help it cycle, and in the meantime the newts are in a large 35l plastic tub with a daily 75% water change. However, the big tank just. will. not. cycle. I gave it a week with no change, and then put the original filter back in the tub, however I must have killed the existing bacteria at some point because there is now a small amount of ammonia in the tub when I test before the water change (still doing daily water changes). Big tank is still stuck on 0.25 ammonia, some nitrites and zero nitrates (has had its own filter, its been about 4 weeks).

So, now what? Keep testing the big tank, and keep the newts in the tub and change the water daily? I'm happy to do that, but they seem stressed - they keep trying to swim up and almost jump out of the water, and spend more time near the surface than they did before. Also, is there any way to speed up a cycle?

I hate cycling tanks, every one I've ever done has taken a different amount of time.
Unless you have access to mature media or decorations from another thank you just need to carry on as you are. You could go out and buy loads of fast growing aquatic plants like anacharis or hornwort, they will help things along nicely, but it still takes time.
here's an article about cycling a tank without any fish/newts in there.

Fishless Aquarium Cycling Method

If you have someone with a healthy pond you can use the pond water to fill the aquarium. Test the quality and check for hazards like dragonfly larvae and such. Watch out for duck weed as well. This can cover the surface pretty quick. Once in your setup it's really hard to get rid of it. Don't put in all your animals at once though.