As far as I know, there is no test to confirm that you are cured of Lyme. Many doctor's believe that once Lyme has become chronic, it cannot be completely eradicated from the body, but rather put into remission.
If you caught the Lyme very early and took an antibiotic right away, it may be possible to prevent it from becoming chronic. However, you stated that you were sick for 2 months which would suggest that the infection had more than enough time to progress to a more serious state.
100 mg of Doxy generally doesn't work to eradicate all three forms of Bb (lyme bacteria). There is also the possibility of co-infections such as bartonella, babesia, erlichia/anaplasma all of which could have been transmitted by a single tick bite.
I would suggest consulting with a LLMD like others have mentioned. He/she can go over your symptoms and help determine whether or not you need additional treatment.