Tenant Smoking like a chimney in a nonsmoking building!

[Nick Taskani]

I bought a 4 unit complex a few months ago, the previous owner had a strict non-smoking policy that was signed by each tenant.  It basically states that smoking constitutes a breach of the lease and grounds for immediate termination of the lease.   I went in today to take some measurements of the windows and sure enough the unit smelled like a chimney.  I saw cigarette butts/ashtrays everywhere.   

Now she is otherwise a good tenant.  Should I give her a warning?  Anyone have experience with this?  If I give her a warning, and if she says she will abide by the rules, do I make unannounced visits and check on her?   

Or should I evict her now?  Seems harsh but I don't want my unit to stink like smoke, hard to get that stuff out.  

Thanks  for any advice.