In Nilsson and Riedels textbook: Electric circuits, it is actually stated on page 28 that " when represented in a circuit diagram, copper or aluminum wiring isn't usually modeled as a resistor; the resistance of the wire is so small compared to the resistance of the other elements in the circuit that we can neglect the wiring resistance to simplify the diagram"
Resitors are poorly conducting, while the wiring in circuit diagrams is typically modeled as a perfect conductor (an equipotential), but of course you are correct, and you can calculate the resistance of a wire using Pouillet's Law as long as you have information about the resistivity of the wire. The formula is:
$$R=\rho\frac{\ell}{A}$$
In Introduction to electrodynamics by Griffiths the resistivity of copper at room temperature is given as $1.68*10^{-8}$ Ohm-m.
Using this information you can form your own conclusions. (Note that $\rho$ varies with temperature)
UPDATE :
John : Thanks for data. Graph is ok. I note your intercept is E=3.94V but your calculations use E=4.5V. This explains the discrepancy in your results. If you use 3.94V you get r ranging from 1.59 to 1.76, close to slope value of 1.68 Ohms.
ORIGINAL ANSWER :
Your line of best fit gives an average internal resistance r based on all measurements. If data points do not lie exactly on this line then the value of r calculated for individual data points (measured pairs of V and I) will not be exactly the same as the slope of the line of best fit.
If you have drawn the line correctly some points will be above the line and some below, with about as many each side, and with the above and below points distributed randomly.
However, it sounds as though there is a consistent trend in your data points : eg all 'below' points at low current and all 'above' points at high current. This suggests that internal resistance was not in fact constant, within the limitations of experimental error. You do not say how big an effect this is : if small, you may be able to ignore it.
EMF and r should be measured when the current drawn is very small, ideally 0. Possibly you have taken readings at a high current, or you have taken a long time to take them. This can have two effects : (i) depleting the battery, reducing EMF, and (ii) increasing r because the battery is warming up and this increases internal resistance.
Your observation that internal resistance increased as current decreased suggests to me that you may have started readings with a high current then worked down to low current.
You will need to decide for yourself what went wrong, perhaps after consulting your teacher again and explaining how you took the readings.
Best Answer
Resistance of a material is $$R=\rho\frac {l}{a} $$ where $\rho $ is the resistivity of the material, $l $ is the length of the conductor, and $a $ is the cross sectional area.
For a given conductor, all these are constant. So resistance of a given conductor does not depend on current or voltage.
It however, depends on temperature: $$R=R_0 (1+\alpha×t) $$ where $R $ is the new resistance, $R_0$ is the initial resistance, $t $ is the change in temperature, and $\alpha $ is the coefficient for increase in resistance per unit rise in temperature.
For copper, resistivity is very low, so resistance is also low.