No real illness dangers. Take the usual precautions.
Adventure? depends how you define that. If you wanted lots of wild outdoor adventure - mountains, hill, river tracing, beaches, as opposed to just ticking off the tourist destinations, you might want to think about renting a vehicle.
Please specify what area. Taipei? kaoushiung? Nantou? Hualien? landscapes vary a lot here. Usual precautions are to be taken: no hiking at night, be aware of snakes in the summer, absolutely NO mountain hiking or river tracing or better yet no outdoors activities during typhoons.
Driving is insane by most standards, but at least you have clearly marked roads and routes and public transportation is readily available, cheap, clean, safe, and reaches most popular destinations, not to mention small busus -most of them fre- that do routes among attractions at popular destinations -like Taitung. Vehicle and local driver is cheap, vehicle driving by yourself is fine and economical enough though.
Kids like the interactive museums here. There are even “factory tours” where kids learn on hands manufacture by themselves, like salt sculpting at the Salt Museum in Tainan, or making animal shaped puppets out of towels at the Towel Factory. There are boat tours of the manglar / wetlands in Tainan which are pretty cool, as well as boat trips at Tamsui and Keelung rivers in Taipei. I guess you could go hiking but a local guide -English speaking tours available, even with foreign guides- would be safer.
Enjoy.
Traffic is probably the most dangerous thing here in Taiwan.
If your kids enjoys the outdoors, there are a lot of hiking, river tracing, water sport options. If you have time, I recommend spend more time in the east coast or southern part of the island.
Taiwan would definitely be a fun and adventurous time for 10 and 12 year olds. The biggest illness danger is currently dengue fever, which is a problem in southern areas of Taiwan such as Kaohsiung. There’s no vaccine for dengue but it’s only serious in a small minority of cases, and because it’s spread by mosquitoes you can use repellent to greatly reduce the small chance of being infected. Maybe by summer 2016 the epidemic will be over. Or you could confine your holiday to non-dengue areas. There’s lots to do just in the north of the island.
Other illnesses you may choose to vaccinate against outside the normal schedule are Japanese encephalitis (extremely remote chance of catching this) and Hepatitis A.
Unlike some Asian countries, the chances of tummy upsets from eating street food etc. are very low.
For food, check out The Hungry Girl’s Guide to Taipei. Hotels depend on your budget. The full range is available in Taipei, but there are fewer of the top of the market hotels outside the capital. If you stay at Taroko Gorge, one of the island’s main tourist attractions, I would recommend the Leader Hotel.
I agree, the Leader Hotel cabins in Taroko are great. Bulowan is wonderful.
But this : [quote]Unlike some Asian countries, the chances of tummy upsets from eating street food etc. are very low.[/quote]
This alas is case-by-case. I eventually grew tired of throwing up all night and came up with a simple solution: no meat on the street! 
Guy
I point it out one more time: Traffic is dangerous. Please pay attention to your kids and make sure they do not assume to have any kind of right of way with any kind of vehicles. Green traffic light for pedestrians and still a truck turns right without looking etc. Avoid getting into arguments with careless drivers even if they are clearly in the wrong. Knocking on a car which almost bumped into you as a pedestrian can get you into a fight with a mouth-foaming driver.
Agree…this is the most dangerous aspect of Taiwan.
EDIT: You and your kids might enjoy a day-trip to Yingge pottery old street. There are some shops there that have hands-on pottery making or simply decorating a pre-made pottery cup. There are also some hiking trails near Yingge. Yingge is extremely easy to get to from Taipei by public transportation.
Some traffic tips for pedestrians:
Do not assume the driver sees you/is aware of you/will stop. My Japanese friends have the bad habit of jaywalking because in Japan the cars will stop for them -yes, but in Japan the metro doors will bounce back open if you stick your hand, not like Taipei that when they close, they close. Cross roads defensively in designated spots, as drivers may not respect even those, and you want the law on your side.
Do not be the first one to cross. Do not run. Be aware that vehicle swill keep moving/turning at zebra crossings in spite of red light. They are supposed to stop for you. Most do.
At the corners, do not stand in the border, or too close to the border, as the buses tend to take sharp turns and you might get a close shave… with a rearview mirror.
Look both ways, even on one way streets.
Sunday drivers are bad -especially on holidays, many people only drive to take their families down South for special holidays/festivals. Worse are blue truck or delivery drivers. Next buses.
Riding buses and taxis is safe. Problem is being newt to them on the road.
Aside from traffic, you are relatively safe from armed robberies, most pickpocket situations, lost items are usually returned to their owners, and generally speaking, it is a safe country compared to most big cities in the West. Traffic problems can be learned to live with a lot of patience and keen observation, and as said, avoiding confrontation.
Also in rural areas you should watch out for tigers. You’ll probably never see one, but at night you may hear them roar.
Afterspivak wrote:
I got that reference!
There are tigers here… but not the big cats. They’re butterflies of the genus Parantica.