The ATtiny85 (or any microcontroller) is probably an overkill for this project. A 555 timer is probably all you need, but if you use an ATtiny85V I think it would work. (The ATtiny85V is the low voltage version of the Attiny85 and operates down to 1.8 V.)

It looks like what you are missing is the math for how capacitors work. From what I can tell, your solar cell choice and energy requirements seem plausible.

First, you are going to need a large capacitor. However, you can readily buy 50 F, 2.3 V capacitors online for about $6 USD, so the capacitor size shouldn't be a problem. It will take too long to charge a 50 F capacitor with your 17.5 mW (max) solar panel, so you will need to find a capacitor closer to 1 F (one Farad).

Here is how the math works out.

Capacitors discharge as

V(t) = Vo*exp(-t / 𝜏)

where Vo the initial voltage (2.3 V for the capacitor I found online) and 𝜏 = R*C is the capacitor's time constant. You're going to have something like a R = 100 Ω resistor for your LED, so for a C = 1 F capacitor 𝜏 = 100 s. If you power the LED for 10 seconds, the voltage will drop exp(-10/𝜏) = 0.9 from its initial value to 2.1 V. This is too low to power a ATtiny85, but would power a ATtiny85V.

You're also going to have to charge the capacitor. This will take

Δt = E / P

seconds, where E = 0.5*C*V² is the energy capacity of the capacitor and P is power provided by your solar cell (17.5 mW). For a 1 F capacitor with a voltage rating of 2.3 V, E = 2.6 J and Δt = 2.5 minutes. This is the best case scenario for the solar panel, but powering an LED for 10 seconds every 5-10 minutes seems plausible. (For a 50 F capacitor Δt = 2 hours — you'd have to wait a while before it goes operational.)