The rate equation for mixed inhibition is v = (Vmax * S)/[Km(1 + i/Kic) + S(1 + i/Kiu)]. Note that there are two Ki values Kic for the competitive and Kiu for the uncompetitive parts of inhibition. You have to fit your data to the equation (non-linear fitting to the original data in simple S-v plot) to get the parameters vor Vmax, Km, Kiu and Kis dependent on substrate (S) and inhibitor (i) concentrations. If the data are good, the lines of an LB plot should (about) meet in one point and you get a good fit with standard 2-variables- fitting programs by separate fittings for each i (look how similar the predicted parameters are in each I series, they should be identical in an optimal case), otherwise you need a program that can handle 3 variables (S, i and v) to give you the 4 parameters.

As for the second substrate, you need to apply the same procedure (initially with less i concentrations. Dependent on the enzyme, there may also be the same type of  inhibition by i for this substrate, or a different type or no inhibition at all (e.g. NAD-dependent dehydrogenases are often inhibited by substrate analogs, which do not affect the reactivity with NAD).

My preferred approach to this problem using steady-state kinetics, if feasible, is to measure the initial rates for a 3-dimensional matrix of substrate A, substrate B, and inhibitor concentration (It helps to have a continuous readout and to use a 96- or 384-well format), then fit the entire data set by nonlinear least squares regression to several different rate equations for the reaction. Each equation is based on the (hopefully) known kinetic mechanism of the enzyme, modified to include terms for the inhibitor: competitive with substrate A, competitive with substrate B, or competitive with both substrates (with a single Ki or two different Kis). Of course, this only works for competitive inhibitors.

I would also do my best to verify, prior to performing this complicated experiment, that the inhibitor is specific. By specific, I mean that it is not an aggregator or a denaturant, that the inhibitor is of high purity, and that the inhibition is not due to a minor but potent contaminant.

Biophysical binding measurements, although they require much more protein than kinetics experiments, can be very helpful in establishing specificity, can also contribute substantively to understanding the mechanism of inhibition, and may also provide another way of measuring Ki. Some examples of biophysical techniques that can yield Ki information include intrinsic tryptophan fluorescence, equilibrium dialysis,the Hummel-Dreyer method, isothermal titration calorimetry, differential scanning calorimetry, surface plasmon resonance, and 2-D protein NMR.

Thank you Johann Heider and Adam B Shapiro for your valuable suggestions.

@Johann Heider: I am trying to calculate according to your suggestion. 

@Adam B Shapiro: We purchased the compound from sigma and confirmed its purity through NMR.

We have checked the binding through intrinsic fluorescence but not using probe (ANS). The problem here is our protein does not have Tryptophan in it. But it possess 8 tyrosine residues and 14 Phenylalanine. Few of them face towards outer surface of protein and few were towards inner surface but none of them are present in the active site.  So far in literature, fluorescence experiments were shown with probe (ANS) only. 

The protein is bisubstrate. When the S1 binds to active site, the protein changes from open to closed conformation (due to closure of the active site loop). Then it accommodates S2. 

We are trying to figure out the binding mechanism of inhibitor.

And we perfomed the fluorescence experiments with protein +inhibitor, protein+S1+inhibitor and protein+S1+S2+inhibitor. With protein alone, we could see the increased quenching with increased concentration of compound. With protein + S1+ inhibitor, more quenching was observed which could be because of S1 as well as inhibitor binding. But, in the presence of S1+S2+Inhibitor, the quenching was decreased (fluorescence intensity was increased). 

My questions are: In this case (without tryptophan), Is the intrinsic fluorescence experiment valid?

If it is valid, how could we understand the influence of S2 on inhibitor binding in the presence of S1? what else could be done?

And we can perform only inhibition kinetics, fluorescence, circular dichroism and FTIR experiments (due to limitations in facility).

Your tyrosine fluorescence quenching experiments sounds promising. They suggest that the inhibitor is competitive with B, and may also be competitive with A. Make sure you control for inner filter effect caused by absorbance of the compound. This can be done by doing the same titration, but replacing the protein with a fluorophore similar to the one in the protein, in this case tyrosine.

When doing tryptophan fluorescence experiments, I use N-acetyltryptophanamide as the control, because of the lack of charge that might result in static quenching with charged substrates. So I would recommend using N-acetyltyrosinamide for your control. It is commercially available (http://www.scbt.com/datasheet-301273.html).

It seems clear from your statement about the enzyme mechanism that it is Ordered Bi. This knowledge will make it easier to do the kinetic analysis, using the rate equations for dead-end inhibition of the Ordered Bi mechanism by A-competitive and B-competitive substrates.

@Adam B Shapiro: Thank you.

It seems that the answer was given in the previous comments. The steady-state formula, derived for a "simple" model is presented above in the answer of J. Heider. It might , however, be extended for other effects, e.g. allosteric, but a simple kinetic analysis cannot reliably detect them, anyhow. Nowadays, the corresponding steady-state and/or equilibrium constants should not be estimated by graphic analysis (slopes, ontercepts etc.). Numerous nonlinear regression software packages allow their computation from data sets containing the two independent variables (concentrations of substrate & inhibitor) which do not need to follow a specific scheme (i.e., they can contain nonsystematic variations of the two), under the condition, that sufficienty enough {s,i} measurings are available. Usually, the iterations of the nonlin routines proceed without problems (simple Newton iteration method is applicable in most of the cases).