Lots of information via search box.
These will process a visa without requiring you to purchase a travel package. Normally 7-10 days but can be done in as little as 4 days. Prices vary per agency and per country passport.
China Travel Service
臺灣中國旅行社股份有限公司
No. 54, Linsen North Road, Zhongshan District, Taipei City, 10491
02 2562 5858
https://goo.gl/maps/Z1kK8sMxduA2
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4 working day processing; HK$1260/US$162/NT$4800
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2 working day processing; HK$1960/US$252/NT$7500; before 9am day 1, return after 3 day 2
STTVisa
勝達簽證通
10450, Taipei City, Zhongshan District, Changchun Rd, 40號6樓之8
02 2541 5271
https://goo.gl/maps/UmSWspbEYsC2
Astor Tours
正達旅行社有限公司
No. 197, Section 3, Nanjing East Road, Zhongshan District, 10491
02 2545 8199