Don't trivialize this because it appears to be minor and inexpensive. If something goes wrong, you could be liable for damage.
You should have an idea of what a reasonable cost should be based on the rent and condition of the property.
Discuss it with the landlord.
If the landlord chooses to fix it, you should have something in writing, signed by both you and the landlord, acknowledging that you aren't responsible for reimbursing the landlord for the cost or acknowledging receipt by the landlord of any contribution from you and a statement that the contribution fulfills your lease requirement.
If you're going to fix it, you should have something in writing, signed by both you and the landlord, agreeing to the choice of replacement parts and the choice of anyone you hire to do the work. It might be nice to think you can save money by doing it yourself. But if anything happens and there's damage, you could be liable.
You should have renter's insurance. You should check your renter's insurance policy to see what, if anything, it says about tenant responsibility for repairs. Your insurance might pay for it. More importantly, will your insurance protect you if something goes wrong and there's damage?
I recommend you copy this and modify it as you see fit or, if you don't this is correct, make up something that you think is correct. Keep an electronic copy on your computer. But also have a notebook in which you keep how-to papers, so you don't have to figure it out all over again at some time in the future and to use the how-tos you accumulate to help you make new how-tos as necessary. Also keep documentation, especially warranties. Scan receipts (receipt ink is notorious for fading over time) and warranties, keep them the electronic copies on your computer and print the receipts and put the printed receipts and original receipts with the warranty papers in the notebook.
Keep a manilla envelope with receipts, warranties, instructions, and any papers you receive from anyone you hire to do the work. When you move out, you should give the envelope to the landlord so the landlord will be able to maintain what was installed and obtain warranty service if necessary.
