Is landlord responsible for getting rid of bed bugs

Yes. Landlords must provide a safe and sanitary living environment to the tenant. This is the implied warranty of habitability. That definitely includes no infestations from things like bedbugs and cockroaches. If the landlord will not fix the issue you should consult with a local landlord/tenant attorney who handles these sorts of disputes. Oftentimes the only remedy though is to move and break the lease.

The tenant probably should not have moved in with furniture from previous tenants still being there. Bedbugs is a tricky issue. I would obtain a statement from a local exterminator showing the reason for the infestation and keep all records of communicating with the landlord in trying to get the situation resolved. If a tenant simply "breaks" the lease, she might still be liable for rent.