It is not unusual for passengers to forget or lose an item while riding in a taxi. In such a case, the first place to check is with the taxi company itself. Unfortunately, passengers do not usually remember exactly which company's taxi they used.
If an item lost anywhere--including taxis--is turned over to the police, there is hope. The police initially give the finder a receipt. Information on that item is then posted to the website of the National Police Agency (see link below). If the owner is found, the owner is required to bring valid ROC ID Card or passport to claim the item. If, after six months, the item isn't claimed, it is given to the finder.
There is also a Police Radio Station that announces lost items, and they have a phone number one can call for assistance. In Taichung, that number is (04) 2251-1911. They are unlikely to speak English, so help from a Chinese friend may be necessary when communicating. Useful Links National Police Agency, Ministry Of The Interior Police Radio Station