How to break a lease without 60 days notice?

In every lease, there is an implied warranty of habitability, which means the landlord must maintain the premises in a habitable condition by complying with local and state housing codes.  The mildew in the ceiling may be a health and safety issue which would constitute a breach of the implied warranty of habitability.  If it is a breach of the implied warranty of habitability, the tenant notifies the landlord as you have done and the landlord is required to respond in a reasonable time by making the necessary repairs.  If the landlord fails to respond within a reasonable time, the tenant has the following remedies for a breach of the implied warranty of habitability: The tenant can make the repairs and deduct the cost from the rent or the tenant can move out and terminate the obligation to pay rent for the balance of the term of the lease or if the tenant stays on the premises, the tenant can withhold rent and defend against eviction.  Another alternative is to sue the landlord for breach of the implied warranty of habitability.

The mildew causing your allergic reaction  would appear to be a breach of the implied warranty of habitability.

Let's say for the sake of discussion that it is not a breach of the implied warranty of habitability and you moved out without giving proper notice, you would only be liable for the rent until the apartment is re-rented (assuming the new tenant is paying the same rent or greater rent than you were paying).  The landlord cannot allow the apartment to remain vacant for the balance of the term of your lease.  The landlord has to mitigate (minimize) damages by making reasonable efforts to find a new tenant.  Reasonable efforts would be based on what other landlords in the area are doing to attract tenants such as posting a sign on the property, advertising in the newspaper or rental guide, etc.  If the landlord did not make reasonable efforts to find another tenant, the landlord's damages would be reduced accordingly.  In other words, the landlord cannot allow the apartment to remain vacant for the balance of the term of your lease to try to hold you liable for the rent for the balance of the term of the lease.  If the landlord finds a new tenant, but the new tenant's rent is less than yours, you would be liable for the difference in rent between what you paid and what the new tenant is paying for the balance of the term of your lease.  Again, the landlord would have to have a valid reason for charging reduced rent such as market conditions; otherwise, the landlord has failed to mitigate damages and the landlord's damages will be reduced accordingly in a lawsuit against you for the rent for the balance of the term of  your lease,