I get nuggets from my mines Sparkling/Deep rarely and chopping rocks as you've mentioned.
You can check the Inventory Guide but a couple suggestions I have:
*Try and have 2 of each, Sparkling and Deep Mine (which can also drop the Mithril you need)
*If you have the Desert Palace level 10, that has a small chance to drop nuggets
*Some Adventures drop them, check Adventure Guide in The Hub (even in this pirate event the 3 hour Adventure had a rare chance of dropping a nugget! So keep an eye on the first page of new events for possible sources)
*Keep tapping on Sir Pigglesworth, he also has a rare chance
This will take a while to complete so just tap away. If you have spare Gems (from game downloads and such) gemming past the pipe requirement might be an option for you.
For Mithril:
*Try and do the Jewelers Exchange as often as possible. It doesn't always drop one but I have had it drop 2 Mithril....twice
*If you have the craft for xmute Silver to Mithril, do that as much as possible. I don't when there's an event that needs the Magic Forge alot though. Try and have 2 Magic Forges for this since it's a 2 day craft.
If you can't xmute Silver yet, then try and tap your Tablet of Tabernum everyday to get the knowledge shard to drop.
My advice, don't give up because you think it's going too slowly.
It was designed that way