It's a 3 part signature both tenant's and you addendum.

They go on to say that if we don't meet and actually provide a signature affirming the condition of the home, that they won't let us show it for the next week as they cannot come back to the home to check the condition after each showing (as if a prospective buyer might take a sledgehammer to the walls??).  

Unless you're in Austin, I bet you can.Next option, which you might want to do either way, is to send them a letter with delivery confirmation (USPS priority mail is great for this, because it doesn't require a signature and they can't avoid proof of delivery since the postman just scans it before they put it in their mailbox) that says that although you have accepted late payments with late fees in the past, going forward you will be enforcing the due date in the lease, including starting eviction proceedings according to law if payments are not paid on time per the lease.Then, when they don't pay on time next month, take the appropriate steps for eviction.And honestly, if it was me, I'd do this anyway.  

Easier to notify, than get signatures.

Automatic Rejection: Your application will be automatically rejected for any of the following:a) Eviction by a prior landlord in the last 24 months;b) Outstanding debt or unpaid damages to a previous landlord;c) Undisclosed criminal record, felony or misdemeanor;d) Being registered as a sex offender;e) Conviction of possession or distribution of a controlled substance;f) Conviction of spousal abuse or violent crime;g) Currently pending the outcome of a felony or misdemeanor charge;h) Bankruptcy in the past 5 years that has not been discharged/released;i) Invalid social security number;j) Outstanding federal tax lien, the repayment terms for which are not current;k) False information on your application;l) Incomplete application, including applicant signature on all pages of the application and this form;m) Failure to pay the application fee at the time of application submittal.

It's a European design signature.

Never send anything requiring a signature. 

When you set up a maintenance inspection, explain that you will be bringing a new agreement to get new signatures.  

If you must require a signature and she just refuses to sign, you can give her proper notice and simply not renew the lease

Next time, also get estoppels with the seller and tenant signatures so you're all on the same page about when rents are due, how much they are, and how much security deposits are owed.Take it as bitter medicine.