Would it be ok to send x-girlfriend of 3 months painting and letter for brithday ?

[Big Paul]

My x-girlfriend and I have been broken up for 1.5 months and I still love her very very much. She broke it off with me because she felt like I wasn't that interested in her, and I was way to distant. We tried the friend thing a few weeks but I came across to desperate to get back with her and it made her run faster. I'm 26 working adult and she is a 21 year old college student. Yesterday after inviting her to lunch she told me "not to text her anymore and thank you for my understanding" I still love her very much and would love to send her a painting she always wanted on June 15 her birthday with a letter asking to reunite friendship with me. I'm so hurt because of the sudden disconnect and when we met I was going through a lot with illness in my family and even starting a new job and she suddenly just disappeared without any reasons. Would it be a good idea to send a gift to her job 2 months from now? My friends say I would of forgotten about her by then, but I doubt it